QUASI PERIODIC ORBITS IN THE VICINITY OF THE SUN-­‐EARTH L2 POINT AND THEIR IMPLEMENTATION IN “SPECTR-­‐RG” & “MILLIMETRON” MISSIONS
IVAN ILIN, ANDREW TUCHIN
KELDYSH INSTITUTE OF APPLIED MATHEMATICS
RUSSIAN ACADEMY OF SCIENCES
65th InternaHonal AstronauHcal Congress
Toronto, 29 September -­‐ 03 October, 2014 APPLICATION OF QUASI PERIODIC ORBITS NEAR LIBRATION POINTS
ESA space telescope “Gaia” launched on 19.12.13, direct transfer to a Lissajous orbit in the vicinity of the Sun-­‐Earth L2 point
Roscosmos space telescope “Spectr-­‐RG” YZ
launch: scheduled at the end of 2016 direct transfer to a Lissajous orbit in the vicinity of the Sun-­‐Earth L2 point
ScienHfic mission: X-­‐rays and Gamma range high precision sky survey, black holes, neutron stars, supernova explosions and galaxy cores study.
Roscosmos space telescope “Millimetron” YZ
2
launch: scheduled in 2019 but may be pos
direct transfer to a big radius quasi-­‐halo orbit in the vicinity of the Sun-­‐Earth L2 point
ScienHfic mission: space observaHon in millimeter, sub-­‐millimeter and infrared ranges. The 12m space telescope will operate at cryogenic temperatures near 4K providing unique sensibility.
3
PERIODIC AND QUASI-­‐PERIODIC MOTIONS AROUND L2 POINT
THREE BODY PROBLEM APPROXIMATION
1.  SoluHon of the system of the linearized equaHons, describing circular restricted TBP
ξ1 = A(t ) cos (ω1t + ϕ1 (t ) ) + C (t ) eλt ,
Amin ≤ A(t) ≤ Amax
ξ2 = −k2 A(t ) sin(ω1t + ϕ1 (t ) ) + k1C (t ) eλt
Bmin ≤ B(t) ≤ Bmax
ξ3 = B (t ) cos (ω2t + ϕ2 (t ) ) ,
C (t) ≤ Cmax
2.  Ricahrdson 3d order analyHcal approximaHon of periodic moHon about the collinear points, obtained with the help of Lindstedt-­‐Poincaree technique applied to Legendre polinomial expansion of the classical CRTBP equaHons of moHon x = a21 Ax2 + a22 Az2 − Ax cosτ 1 + (a23 Ax2 − a24 Az2 )cos2τ 1 + (a31 Ax3 − a32 Ax Az2 )cos3τ 1
y = kAx sinτ 1 + (b21 Ax2 − b22 Az2 )sin2τ 1 + (b31 Ax3 − b32 Ax Az2 )sin3τ 1
z = δ n Az cosτ 1 + δ nd21 Ax Az (cos2τ 1 − 3) + δ n (d32 Az Ax2 − d31 Az3 )cos3τ 1
Az ≥ 0
Ax > 0
Ax min ≥ Q
4
δ n = 2 − n, n = 1,3
QUASI PERIODIC SOLUTION IN RTBP TRANSITION FROM CIRCULAR RTBP TO ELLIPTIC RTBP
CRTBP
ERTBP
• Quasi-­‐periodic orbit approximaHon: • LibraHon points in the RTBP are homographic, which means they keep their relaHve posiHon when transfer to the ERTBP is performed
Richardson model
r
X(t)
• State vector , lying on the obtained • Transfer from dimensionless true anomaly f , describing evoluHon of the ellipHc system, to the dimensionless Hme t of the TBP is performed
quasi periodic soluHon
ρ=
p
(1 + e cos f )
dx dx dt
= ⋅
df dt df
dt
p3 2
=
df (1 + e cos f )2
ξ ! = x!
32
⎛ f ⎞
tg ⎜ ⎟
2
⎛ E ⎞
tg ⎜ ⎟ = ⎝ ⎠
1+e
⎝ 2 ⎠
1−e
M = E − e sinE
tdimensionless = M
• A periodic halo orbit approximaHon is built with p
+ x (−e sinf )
(1+ e cosf )
the help of Richardson model
p3 2
η! = y!
+ y (−e sinf )
(1+ e cosf )
• TBP state vector (x ,y ,z , x! ,y! ,z!) is converted to p3 2
ζ ! = z!
+ z (−e sinf )
(1+ e cosf )
Nehvill dimensionless variables (ξ ,η ,ζ ,ξ! ,η! ,ζ!)
5
x = ρξ
y = ρη
z = ρζ
TRANSFER TRAJECTORY DESIGN -­‐ THE ISOLINE METHOD
The transfer trajectory to the selected quasi-­‐
periodic orbit is searched within the invariant manifold of the restricted three body problem with help of the isoline method. This method provides connecHon between periodic orbit dots and geocentric transfer trajectory parameters – the isoloines of transfer trajectory pericentre height funcHon depending on periodic orbit parameters are built. This provides one-­‐impulse transfer from LEO to the quasi-­‐periodic orbit.
!
x (x ,y ,z , x! ,y! ,z!)
17
⋅ rL
24
!
ξ (ξ ,η ,ζ ,ξ! ,η! ,ζ!)
Earth
rL
L2
rπ , rα , i, Ω, ω, τ
Periodic orbit parameters:
A, B, C, D, φ1, φ2
1-­‐impulse flight trajectories are separated out with the condiHon: rπ = rπ *
With the fixed A, B and C = 0 the isoline is built in the φ1, φ2 plane:
rπ (φ1 ,φ2 ) = rπ *
6
LEO parameters:
SPACECRAFT TRAJECTORY CALCULATION ALGORITHM STRUCTURE
Quasi-­‐periodic soluHon obtained in ERTBP
Data, inclinaHon selecHon
Isoline building algorithm
SelecHon of trajectories with min ΔV
Shadow zones and radio visibility zones calculaHon, soluHon analysis
StaHonkeeping impulses calculaHon
Transfer trajectory iniHal approximaHon building algorithm Edge condiHons: correcHon of B value, C = 0; max t in the vicinity of periodic soluHon
Transfer trajectory exact calculaHon 7
Periodic orbit parameters A, B, LEO parameters rπ , rα STATIONKEEPING IMPULSES CALCULATION
The correcHon impulse vector is calculated according to the condiHon of the maximum Hme of the spacecraq staying in the L2 point vicinity of the stated radius aqer the correcHon has been implemented. The maximum Hme is searched for with the help of the gradient method. r 1 ΔV max
T
ΔVi = q
(∇Fc )
2 ∇Fc
q -­‐ step decrease controlling coefficient.
R (θ A ,θB ) = rL ⋅ θB2 + 1 + k22 θ A2
(
)
⎧ toutL2 − tinL2
⎪
FC = ⎨ 1 t1 +T
2
2
B
t
−
θ
r
+
C
t
( ) dt
⎪ T ∫ ( ( ) B L )
⎩ t1
(
)
8
ΔVmax -­‐ the biggest possible value of the impulse;
THE ISOLINE METHOD FOR THE MOON SWING BY TRANSFERS Advantages: Moon swing by maneuver allows to obtain more compact orbit
Disadvantages: Trajectories become Hme-­‐dependent and mission robustness decreases as the ΔV
maneuver execuHon errors may cost a lot of to correct
1
x 10
9
X-Y
1
50
0.8
60
x 10
9
X-Y
0.8
40
70
0.6
0.6
30
80
0.4
6070
20
0.4
80
50
90
40
90
0.2
10
28010030
460
0.2
4
0
4
20
1
460
10
1
0
280 100
-0.2
-0.2
-0.4
-0.4
-0.6
150
-0.6
150
-0.8
-2
0
2
4
6
8
10
12
14
16
x 10
8
Quasi-­‐periodic orbit obtained without the Moon swing by maneuver
Ay = 0.8 million km
-1
-4
-2
0
2
4
6
8
10
12
14
16
x 10
8
Quasi-­‐periodic orbit obtained with the help of Moon swing by maneuver
Ay = 0.5 million km
The XY plane view of the rotaHng reference frame, mln km.
9
-1
-4
-0.8
ΔV
Total costs are less than 15 m/s for 7 years period
.
10
QUASI PERIODIC ORBITS FOR “SPECTR-­‐RG” & “MILLIMETRON” MISSIONS
QUASI PERIODIC ORBITS FOR “SPECTR-­‐RG” & “MILLIMETRON” MISSIONS XY, XZ, YZ PROJECTIONS ON THE ROTATING REFERENCE FRAME
-­‐1000
1000
-­‐200
-­‐600
1000
-­‐200 -­‐600
-­‐1000
Dimension: thousands of km
11
1000
EVOLUTION OF DIMENSIONLESS PARAMETERS DESCRIBING THE QUASI-­‐PERIODIC ORBIT GEOMETRY
A
RL
θB =
B
RL
θB
θС =
С
RL
θA
θC
t , days
12
θA =
MISSIONS’ CONSTRAINTS
Mission
Spectr-RG
Maneuver
Av, m/sec
Max, m/sec
Av, m/sec
Max, m/sec
1st MCC
2nd MCC
3d MCC
4th MCC
Stationkeeping ΔV costs
22.2
2.4
0.3
1.2
35.0
43.8
10.1
1.3
2.8
124.0
22.2
3.3
0.4
1.4
43.0
43.8
13.5
1.6
3.0
173.0
Total ΔV costs
61.1
182.0
70.3
234.9
Requirements/ Constraint
Driver(s)
Sp-RG: Communications
Millimetron: Science
Spectr-RG
Millimetron
Maximum
900.000 km
Minimum
900.000 km
Orbit geometry: Z amplitude
Sp-RG: Communications
Millimetron: Science
Maximum
600.000 km
Minimum
900.000 km
Maximum SCE finite-burn duration
Minimum precision of SCE finiteburn duration
Estimated MCC average ΔV
Stationkeeping available ΔV
Mission lifetime goal
Lunar / Earth Eclipse
Radio visibility
Propulsion
Propulsion
1800.0 sec
0.1 sec
2000.0 sec
0.2 sec
Mass & Propulsion
Mass & Propulsion
Science
Power and Thermal
Communications, navigation
& control
55 m/s
228 m/s
7.5 years
None allowed
Must be provided
every day for
Northern hemisphere
ground stations
28 m/s
287 m/s
7.5 years
None allowed
Southern
hemisphere ground
stations should be
used
Orbit geometry: Y amplitude
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Quasi-periodic orbit parameters
Millimetron
SPECTR-­‐RG SOLUTIONS MAP
14
1879 soluHons
126 days
904 soluHons
84 days
15
MILLIMETRON SOLUTIONS MAP
RESEARCH RESULTS
§  A new method of quasi periodic orbits construcHon, generalizing Lindstedt-­‐
Poincaree-­‐Richardson technique for the ERTBP case has been developed and programmed.
§  M.L. Lidov’s isoline building method providing one-­‐impulse transfers from LEO to a quasi-­‐periodic orbit in the vicinity of a collinear libraHon point has been extend on gravity assist trajectory class.
§  An algorithm calculaHng staHonkeeping impulses for the quasi periodic orbit maintainance has been developed and programmed. It provides staHonkeeping ΔV
strategies for spacecraq lifeHme over 7 years, total costs are within 15 m/sec.
§  Nominal trajectories for Spectr-­‐RG and Millimetron missions have been obtained by performing the calculaHon described above in the full Solar system ballisHc model. All the restricHons such as Earth and Moon shadow avoidance condiHons and constant radio visibility from the Northern hemisphere have been met.
16
§  Engine error modeling has been performed, the most pessimisHc scenario results saHsfy both mission’s constraints.