The ballistic support of
the “SPECTR-RG” spacecraft flight to the
L2 point of the Sun-Earth system
I.S. Ilin, G.S. Zaslavskiy, S.M. Lavrenov, V.V. Sazonov,
V.A. Stepaniants, A.G. Tuchin, D.A. Tuchin,
V.S. Yaroshevskiy
Keldysh Institute if Applied Mathematics RAS
2012
The quasi-periodic orbits in the vicinity of the
L2 point of the Sun-Earth system
X-Y-Z
100
x 10
8
1
70
6080
X-Y-Z
50
90
0.5
46040
280 100
400
x 10
0
8
30
0
1
20
Z
4
4
10
3
-0.5
2
1
60
70
-1
80
1000
90
50
0
460
280100
Z
40
-1
1
10
20
-150
-2
-3
9
0
12
0
8
-0.5
16
4
12
2
x 10
-1000
Y
6
8
0
8
14
10
6
-2
-4
2
-6
-8
Y
-800
-2
4
x 10
0
-4
X
-400
1600
8
14
10
6
150
-5
10
8
-500
16
0.5
x 10
-4
1000
150
-1.5
1
4
30
2
-1
-2
4
x 10
0
-4
-400
X
8
1600
Missions to the L2 point of the Sun-Earth
system
• Two Russian missions are to be sent to the vicinity of the L2
point during the next few years:
• The «Spectr-RG» spacecraft, flying to the L2 point of the
Sun-Earth system and staying at the halo orbit in it’s
vicinity. NPO S.A. Lavochkina, 2015.
• The «Millimetron» spacecraft, flying to the L2 point of
the Sun-Earth system and staying at the halo orbit in it’s
vicinity. The spacecraft has to go out far from the ecliptics
plane. NPO S.A. Lavochkina, 2018.
• The examples of the L2 point missions that have already been implemented:
• NASA spacecraft «WMAP», (2001 – 2009)
• ESA spacecraft «Planck» + space observatory «Hershel» (2009)
• ESA space observatory – spacecraft «Gaia» should go to the vicinity of the
L2 point of the Sun-Earth system in 2013
The «Spectr-RG» mission
• The «Spectr-RG» mission presupposes the flight to the vicinity
of the Sun-Earth system L2 point and the halo orbit motion in
the L2 point vicinity during the 7 years period.
• The halo orbit in the vicinity of the Sun-Earth system L2 point
is opportune because of the possibility of reaching it with a
single-impulse flight with no correction at it’s end.
• To keep the spacecraft in the halo orbit the stationkeeping is
needed. Total stationkeeping costs for the 7 years period must
not overcome 200 m/sec.
The isoline of the pericentre height function
building method.
• The isoline method for the approximate description of the Earth – L2
trajectories was suggested in M.L. Lidov’s papers. It was applied for the
direct single-impulse flights without any Lunar swing by maneuver.
• The spacecraft motion is described in the rotating reference frames: in the
geocentric reference frame Ox1x 2 x 3 and in the reference frame with the
beginning in the L2 libration point O1 23
ξ3
The average values of A(t) и B(t) are
chosen at the halo orbit designing
stage.
The average value of C(t) must be
close to 0.
ξ2
L2
ξ1
x3
x2
Earth
x1
The
direction to
the Sun
The linearized equations of the spacecraft motion
in the quasi-periodic orbit in the rotating reference
frame
1  A  t  cos 1t  1  t    C  t  et  D t  et
2  k2 A  t  sin 1t  1  t    k1 C  t  et  D t  et 
3  B t  cos 2t  2 t 
The integration constants
 9B
1  n1 
1
2
1  n1 
1
2

 8BL  BL  2  0.035384
2
L
 9B
2
L

 8BL  BL  2  0.042734
2




1

   2 BL  1  0.54525;
k1 
2   / n1   n1 



 
1
1  
;
rad
day
2  n1  BL  0.034148
rad
day
   2

1
 1   2 BL  1  3.1873;
k2 
2 1 / n1   n1 



 1    3
BL   3  3  a1 ;
 r
rL 
 L1
µ1, µ – the Sun and the Earth
gravitational constants
a1 – the astronomical unit;
rad
day
rL1, rL – the distances from the L2
point to the Sun and the Earth;
n1 – the average angle speed of the
Earth orbital motion.
The isoline building algorithm
•
The search of the pericentre height function
f A ,B ,1,2  according to the following algorithm:
1.
2.
3.
•
•
The spacecraft state vector is calculated in the inertial reference frame,
obtained by the fixation of the rotating reference frame axes at a fixed
moment of time according to the parameters: А, B, 1 and  2 .
The obtained state vector is converted into the non-rotating geocentric
ecliptic reference frame.
The geocentric orbit elements are counted with the help of the obtained
state vector with the pericentre height r among them.
The first isoline dot search
The extension of the isoline to the next dot
The first isoline dot search
The scanning is performed within the interval from 0 to 360° for φ1 and within
the interval from –180° to 180° for φ2 with the step of 45º for φ2 and 1º for φ1 .
The φ1 value satisfying the following condition is looked for:
 r 1 1,2   r*    r 1,2   r*   0


With the help of the bisection method the φ1m value satisfying
the following condition is searched :


r 1m , 2  r*  
The pair of φ1m, φ2 values found is the isoline beginning point.
The extension of the isoline from the current point
φ2
1b  1i  s,
 2 b   2 i  s,
φ1b, φ2b
φ1i+1, φ2i+1
φ1i, φ2i
φ1i-1, φ2i-1
 h, если i  1;

h
s
, если i  1
      2      2
1i
1i 1
2i
2 i 1

φ1
The examples of the obtained isolines
The isolines within the
18.12.14 launch window
with the Moon swing by
maneuver and 1 lap at the
LEO
The isolines within the
27.01.14 launch window
with the Moon swing by
maneuver
φ2
φ2
A  0.14, B  0.1
The isolines without the
Moon swing by maneuver
φ2
φ
φ
φ
1
1
1
A  0.12, B  0.1
 A from 0.18 to 0.2.
 B = 0.1
The structure of the nominal transfer trajectory
calculation algorithm
• The isolines built are the income data for the flight trajectory
initial kinematics' parameters calculation algorithm – the initial
approximation of the transfer to the halo orbit.
• The initial approximation built is used for the exact calculation
of the flight from the Earth orbit with the fixed height to the
given halo orbit. The kinematics' parameters vector is counted
more precisely according to the edge conditions.
• The velocity impulses, needed for the stationkeeping of the
spacecraft in the given area around L2 point are counted.
• The shadow zones and radiovisibility zones for the locating
stations, situated on Russian territory are counted for the
whole spacecraft lifetime.
The initial approximation calculation. The transition
from the transfer trajectory to the halo-orbit
1,  2 , 3 , 1,  2 , 3
17
 rL
24
Earth
rL
LEO parameters:
Halo-orbit parameters:
rπ , rα , i, Ω, ω, τ
A, B, C, D, φ1, φ2
The condition to select the one impulse transfer trajectories:
With the fixed A, B и C = 0 an isoline is build in the φ1,
φ2 plane:
r 1, 2   r*
r  r*
L2
The stages of the nominal trajectory calculation
1.
The velocity vector of the hyperbolic transfer trajectory,
obtained from the initial approximation is counted more
precisely according to the edge conditions which are the
given values of the parameters B and C = 0.
2.
The velocity vector, obtained at the stage 1 is counted more
precisely according to the condition of the maximum time of
the halo-orbit staying in the L2 area of the following radius:
RL2  rL  B2   A2 1  k22 
The calculation of the stationkeeping impulses,
keeping the spacecraft in the halo orbit in the L2 area


R  A ,  B   rL   B2  1  k22  A2
,
,
 toutL2  tinL2

FC   1 t1 T
2
2
B
t


r

C
t
dt






B L
T 
 t1
 FC

(i)
 Vx
 Vx 
 F

 1
Vmax
(i)
 Vy  = k 
 C
2
2
2  Vy

 2



F

F

F
(i)




 V 
C
C
   C   F
 z 

  
 Vx   Vy   Vz   C
 Vz
FC FC FC
,
,
Vx Vy Vz












- the partial derivatives of the FC function with respect to
the components of the velocity vector
Vmax
- the biggest possible value of the impulse;
k
- the coefficient, controlling the step decrease.
The isoline method for the Moon swing by
transfers
It is opportune to use a Moon swing by maneuver for the halo
orbit transfer trajectories, as it allows to find the orbits coming
closer to the L2 point.
For calculations of the pericentre height corresponding to the given halo orbit
the trajectory is divided into 3 parts:
• the flight from Earth to the entrance into the Moon incidence sphere,
• the flight inside the Moon’s incidence sphere,
• the flight after leaving the Moon’s incidence sphere till the entrance of the L2
point vicinity.
For searching the pericentre height these parts of trajectory are passed
backwards. The function of the pericentre height also depends on time in case of
the Moon swing by maneuver being applied.
The transfer trajectory without the Moon swing
by maneuver
1
x 10
9
X-Y
50
0.8
60
40
70
0.6
30
80
0.4
20
90
0.2
10
4
1
0
460
280 100
-0.2
-0.4
-0.6
150
-0.8
-1
-4
-2
0
2
4
6
8
10
12
14
16
x 10
8
The XY plane view, the rotating reference frame, mln. km.
The transfer trajectory with the Moon swing by
maneuver
1
x 10
9
X-Y
0.8
0.6
6070
0.4
80
50
90
40
28010030
460
0.2
4
20
10
1
0
-0.2
-0.4
150
-0.6
-0.8
-1
-4
-2
0
2
4
6
8
10
12
14
16
x 10
8
The XY plane view, the rotating reference frame, mln. km.
The transfer trajectory with the Moon swing by
maneuver and the preliminary lap at the LEO
1
x 10
9
X-Y
0.8
0.6
0.4
60 50
70
40
30
20
80
0.2
90
4
10
280 100
0
1
460
-0.2
150
-0.4
-0.6
-0.8
-1
-4
-2
0
2
4
6
8
10
12
14
16
x 10
8
The XY plane view, the rotating reference frame, mln. km.
The XY, XZ, YZ plane views of the halo-orbit in the
rotating reference frame. The transfer to the halo-orbit is
performed with the help of the Moon swing by maneuver
Dimension: thousands of km
-200
200
200
500
1500
-200
1500
-500
The total characteristic velocity costs for the stationkeeping are about 30 m/sec
for the 7 years period.
500
The XY, XZ, YZ plane views of the halo-orbit in the rotating reference
frame. The transfer to the halo-orbit is performed with the help of the
Moon swing by maneuver. There was 1 preliminary lap at the LEO.
Dimension: thousands of km
-200
200
200
500
1500
-200
1500
-500
The total characteristic velocity costs for the stationkeeping are about 30 m/sec
for the 7 years period.
500
The halo-orbit, calculated for the «Millimetron» project.
The XY, XZ, YZ plane views in the rotating reference
frame
Dimension: thousands of km
900
1100
-1100
900
1500
-700
1500
-1100
The total characteristic velocity costs for the stationkeeping are about 14 m/sec
for the 7 years period.
1500
The evolution of the orbit parameters  A , B and  C
A 
A
RL
B 
B
RL
C 
C
RL
B
A
C
t , days
The transfer to the L2 vicinity with the help of the Moon swing by maneuver
The dates of the transition to the L2 vicinity for 2014 year
Month
January
February
March
April
May
June
July
August
September
October
November
December
θA
Launch date
The duration of the launch window, hours
0.14
0.15
0.14
0.15
0.12
0.12
0.12
0.13
0.14
0.15
0.12
0.13
0.14
0.15
0.15
0.14
0.15
0.12
0.12
0.12
0.12
20140128
20140128
20140227
20140226
20140329
20140427
20140529
20140529
20140529
20140529
20140625
20140625
20140625
20140625
20140725
20140824
20140823
20140922
20141023
20141121
20141218
36
72
40
48
46
24
20
28
36
52
22
33.5
40.5
60
41
14.5
57.5
12.5
6
23.5
22
The contingencies for the «Specter-RG»
spacecraft orbit
• To provide the needed level of solar cell panels luminance and
radiovisibilty conditions for the Russian tracking stations, the
following circumstances were taken into account at the orbit
design stage:
• If the spacecraft comes too close to the ecliptic plane,
the penumbra area entrance is possible;
• If the spacecraft goes too far from the ecliptic plane,
long periods of no radiovisibility are highly probable.
The results of the research
• The ballistic problem of obtaining halo orbits with the given
geometric dimensions in the ecliptic plane and in the plane
orthogonal to it has been solved.
• A new method of transfer trajectories building for the flight
from LEO to the family of halo orbits in the vicinity of the
Sun-Earth system L2 point is developed. These trajectories
need no impulse for the transfer from the flight trajectory to
the halo-orbit.
• The stationkeeping velocity costs are evaluated.
• The primary evaluations of the orbit parameters determination
and the forecast accuracy have been obtained.