1. Deployment Optimization for Various Gravitational Wave Missions
An-Ming Wu
National Space Organization, Hsinchu City, Taiwan
Wei-Tou Ni
National Tsing Hua University, Hsinchu City, Taiwan
Gang Wang
Gran Sasso Science Institute (INFN), L’Aquila, Italy

2. Outline Introduction Deployment of ASTROD-GW Formation
Deployment of LISA Formation
Conclusion

3. Introduction
Space Gravitational Wave (GW) mission proposals often use constellation or formation of Earthlike orbits around the Sun.
LISA has three spacecraft in a nearly equilateral triangle formation with 2.5 Mkm arms, inclined with respect to the ecliptic by 60˚, and trailing Earth by 20˚.
TAIJI is proposed to have 3 Mkm arm-length LISA-like orbits.
ASTROD-GW has 3 spacecraft near Lagrange points with arm length about 260 Mkm.
Since the formation deployment is related to the spacecraft mass and trransfer time, it is critical for the mission cost.

4. ASTROD-GW Orbit Configuration with Inclination

5. ASTROD-GW Mission Orbit Parameters
Element
Spacecraft 1
Spacecraft 2
Spacecraft 3
Semi-Major Axis
1 AU
Eccentricity
Inclination
1~3 deg
Right Ascension of the Ascending Node
270 deg
plus that of Earth
30 deg
150 deg
Argument of Perihelion
0 deg
True Anomaly

6. Delta-Vs to escape from GEO and to enter mission orbit
S/C (Destination)
Transfer Orbit
Delta-V from GEO to Inclined Solar Orbit from GEO [km/s]
Delta-V to Mission Orbit from Hohmann Transfer Orbit
[km/s] 1
(near L3)
2 rev in yr
2.570
3.335
3 rev in yr
1.915
1.991 2
(near L4)
1 rev in yr
1.916
1.992
2 rev in yr
1.607
0.903 3
(near L5)
1 rev in yr
1.740
1.422
2 rev in yr
1.594
0.765

7. LISA-Like Formations Spacecraft 1 d ai Reference Object ae
60 ae ai
Spacecraft 1
Spacecraft 3
Spacecraft 2
Reference
Object

8. Orbit Parameters of LISA-Like Formation Armlength = 6 Mkm
Element
Spacecraft 1
Spacecraft 2
Spacecraft 3
Semi-Major Axis
1 AU
Eccentricity
.01158
Inclination
1.149 deg
Right Ascension of the Ascending Node
0 deg
plus that of Earth plus 260 deg
120 deg
240 deg
Argument of Perihelion
-90. deg
Mean Anomaly
180. deg
60. deg
-60 deg

9. Orbit Maneuvers Eccentricity change at periapsis or apoapsis
Inclination change at the ascending or descending nodes
Combination of changes at nodes
Escape from Earth and along the track

10. Compact Finite-Difference Method for Orbit Equation
LHS
Acceleration
Compact Finite Differencing
Accurate with 4th Order
RHS
Forces
Implicit Newton Method
Robust with Incresing Diagonal Dominance

11. Transfer Orbits of 3 LISA-like Spacecraft from LEO to Mission Orbits with Transfer Time of 180 day

12. Delta-Vs for LISA-like spacecraft with arm length of 6 MkmSC
Angle at LEO
Delta-V from LEO
Delta-V to Mission Orbit 1
270 deg
km/s
2.973 km/s 2
km/s
1.814 km/s 3
9.137 km/s
2.199 km/s

13. Mean Delta-Vs Arm Length of Formatiom (Mkm)
Mean Delta-V to Mission Orbit (km/s)
1.059 1
1.193 2
1.406 3
1.630 4
1.857 5
2.090 6
2.329

14. New LISA Mission
LISA is proposed to be launched around with transfer time of 400 day to arrive mission orbits.
Assume that LISA will be launched on

15. LISA Animation

16. LISA Animation

17. LISA Animation

18. LISA Formation

19. SC1 Deployment Initial Final Earth SC1 Initial SC1 Middle of
Ascending and Final
Descending
Ascending
Middle of
Initial and Ascending
Periapsis

20. SC2 Deployment Initial Final Earth SC1 Initial SC1 Middle of
Initial and Ascending
Ascending
Periapsis
Descending
Middle of
Ascending and Final

21. SC3 Deployment Descending Initial Final Earth SC1 Initial SC1
Periapsis
Middle of
Initial and Ascending
Middle of
Ascending and Final
Ascending

22. Delta-Vs in LISA Proposal
Transfer Time = 400 day

23. Delta-Vs of Presented Calculation

24. Conclusion
We use a stable 4th-order compact finite-difference method to calculate the delta-Vs for the deployment of various LISA-like formations for fixed travel time transfer from LEO.
For quick deployment, the delta-Vs of three spacecraft are different, the mean delta-V is nearly constant for different configurations of the formation, and linearly related to the arm length.
To minimize delta-Vs, the orbit maneuver for combination of eccentricity and inclination is studied.