Deployment Optimization for Various Gravitational Wave Missions An-Ming Wu (amwu@nspo.narl.org.tw) National Space Organization, Hsinchu City, Taiwan Wei-Tou Ni (weitou@gmail.com) National Tsing Hua University, Hsinchu City, Taiwan Gang Wang (gwanggw@gmail.com) Gran Sasso Science Institute (INFN), L’Aquila, Italy 2017.5.23

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

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.

ASTROD-GW Orbit Configuration with Inclination

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

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 1.500 yr 2.570 3.335 3 rev in 2.500 yr 1.915 1.991 2 (near L4) 1 rev in 0.833 yr 1.916 1.992 2 rev in 1.833 yr 1.607 0.903 3 (near L5) 1 rev in 1.167 yr 1.740 1.422 2 rev in 2.167 yr 1.594 0.765

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

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

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

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

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

Delta-Vs for LISA-like spacecraft with arm length of 6 Mkm SC Angle at LEO Delta-V from LEO Delta-V to Mission Orbit 1 270 deg 11.297 km/s 2.973 km/s 2 10.213 km/s 1.814 km/s 3 9.137 km/s 2.199 km/s

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

New LISA Mission LISA is proposed to be launched around 2028-30 with transfer time of 400 day to arrive mission orbits. Assume that LISA will be launched on 2030.12.18.

LISA Animation

LISA Animation

LISA Animation

LISA Formation

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

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

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

Delta-Vs in LISA Proposal Transfer Time = 400 day

Delta-Vs of Presented Calculation

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.