Final Version Steven Cooley Rich Luquette Greg Marr Scott Starin Flight Dynamics May 13-17, 2002 Micro-Arcsecond X-ray Imaging Mission, Pathfinder (MAXIM-PF)

Final Version Flight Dynamics Page 2 MAXIM-PF, May 13-17, 2002 Goddard Space Flight Center Requirements & Assumptions (1 of 2)  Phase km +/- 5 m 5cm control 15  m Knowledge  Phase 2 5cm control 15  m Knowledge Optics Hub S/C Detector S/C 20,000 km +/- 5 m FreeFlyer S/C m separation Control to ~10 microns Detector S/C Optics Hub S/C

Final Version Flight Dynamics Page 3 MAXIM-PF, May 13-17, 2002 Goddard Space Flight Center  Mission Orbit  L2 Lissajous  Heliocentric “Drift-Away”  Variations on Drift Away (e.g., DROs stay closer to Earth)  Orbit Control and Knowledge Requirements  Orders of Magnitude above Current Operational Missions  Not Addressed Here  V and Acceleration Magnitude Values  Very Coarse Approximations  No Noise  CRTBP or Free Space Model  No Perturbations (Moon, Jupiter, etc.)  No Navigation Errors  Further Analysis Required Requirements & Assumptions (2 of 2)

Final Version Flight Dynamics Page 4 MAXIM-PF, May 13-17, 2002 Goddard Space Flight Center Lissajous Orbit Option  Orbit Characteristics  Quasi Orbit Period of ~6 months  Can Choose small or Large Amplitude Lissajous  No Earth Eclipses  MAXIM Adds Requirement of No Lunar Shadows (MAP)  Advantages  Spacecraft do not Drift too Far from Earth  Communications (High Data Rate Missions)  Spacecraft can be More Easily Replaced/Repaired  Important for Long Missions  Small Launch Vehicle C3 (-2.6 for Phasing Loops, -0.7 for Direct)  Disadvantages  Unstable Complicated Dynamics  Can Lose Spacecraft (e.g., Propulsion Failure)  All s/c in formation require propulsion (Operational Complexity)  Formation Keeping Costs May be Greater (Further Analysis Needed)  May Have increased variation in Formation Keeping Control Acceleration Magnitude (Harder to size thrusters)  6 Month Transfer Time  High Thrust Propulsion System Likely Needed (Need to Correct LV Errors QUICKLY)

Final Version Flight Dynamics Page 5 MAXIM-PF, May 13-17, 2002 Goddard Space Flight Center Heliocentric Orbit Option  Orbit Characteristics  Drift Away Orbit (0.1 AU/year)  Advantages  Stable Dynamics  Simpler Operations  Potentially No Orbit Overhead Costs  Optics Hub may Not need propulsion  Relatively Short Transfer Times  May Require Less Formation Keeping Costs (?)  May be Able to Eliminate Need for High Thrust Propulsion System  Disadvantages  Higher Launch Vehicle C3 (0.4)  Drift Away Concerns

Final Version Flight Dynamics Page 6 MAXIM-PF, May 13-17, 2002 Goddard Space Flight Center Lissajous Orbit Option Phase 1

Final Version Flight Dynamics Page 7 MAXIM-PF, May 13-17, 2002 Goddard Space Flight Center  Desired Characteristics  Two S/C in formation, 200 km apart  Maintain inertial orientation of SC-to-SC line for 1 week observation  Optics Hub follows a ‘Ballistic’ lissajous orbit during Observation (the “Leader”)  Detector SC (the “Follower”) follows a shifted trajectory  For Given Observation, Position differs by a constant baseline vector  Driving Requirements  Time allocated for reorienting the SC-to-SC line  SC-to-SC line remains inertially fixed during observation Lissajous Orbit Description (Phase 1)

Final Version Flight Dynamics Page 8 MAXIM-PF, May 13-17, 2002 Goddard Space Flight Center Formation Initialization  Direct Transfer (One LV with a C3 of -0.7 km 2 /s 2 )  Large ‘Halo’ Orbit  No Lunar Shadows  Max L2-Earth-Vehicle Angle  30   Orbit Does Not “Collapse”  Detector SC is maneuvered to the shifted orbit 200 km away  Consider Initialization  V as 6 Formation Re-Orientations  FreeFlyers Stay Attached to Optics Hub

Final Version Flight Dynamics Page 9 MAXIM-PF, May 13-17, 2002 Goddard Space Flight Center Formation Keeping Solar Radiation Pressure (SRP)  Acceleration Magnitude (1 AU)  4.5 x (1+r) A/M (m/s 2 )  A = Cross Sectional Area exposed to Sun (m 2 )  M = Mass of Spacecraft (kg)  r = Reflection Factor. (r  [0,1])  Approximate Result for all Mission Orbits Considered  SMAD (3 rd Edition, not 2 nd edition)  SRP Acceleration Magnitude Differential Between 2 Spacecraft  4.5 x | (1+r1) (A1/m1) – (1+r2) (A2/m2)|  Assumed Dominant Term for 200 km Baseline (CRTBP model)  Assume Control Acceleration Magnitude  m/s 2 Needed

Final Version Flight Dynamics Page 10 MAXIM-PF, May 13-17, 2002 Goddard Space Flight Center Formation Re-Orientation Free Space Analysis (1/4)  Preliminary “Drift-Away” Orbit Results  For “small” reorientation times (< 1 week), solar gravity has “small” effect on  V costs.  Free space analysis (ie, gravity free) is a reasonable approximation for small reorientation times in a “Drift Away”  Further Study Needed (Especially for Applicability to Lissajous Orbits) Optics Hub 200 km 10  Distance Detector at Obs 1 Detector at Obs 2

Final Version Flight Dynamics Page 11 MAXIM-PF, May 13-17, 2002 Goddard Space Flight Center Formation Re-Orientation Free Space Analysis (2/4)  Impulsive Burn Analysis  One burn after obs 1 initiates translation of detector to the obs 2 location  Magnitude:  V Impulse = distance / reorientation time  Equal but opposite burn stops translation when obs 2 location is reached  Total  V = 2*  V Impulse  Continuous Thrust Analysis  Acceleration is constant toward obs2 location for first half of the time  Acceleration is of the same magnitude, but reversed for the remaining time  Total  V (m/s) = 4*  V impulse  Acceleration = 4*  V impulse / reorientation time = 4*distance / (reorientation time) 2

Final Version Flight Dynamics Page 12 MAXIM-PF, May 13-17, 2002 Goddard Space Flight Center Formation Re-Orientation Free Space Analysis (3/4)  V Costs (both Continuous and Impulsive)  Linear Relationship with Distance  Inverse Linear Relationship with Re-Orientation Time  Control Acceleration Magnitude (Continuous)  Linear Relationship with Distance  Inverse Square Relationship with Re-Orientation Time

Final Version Flight Dynamics Page 13 MAXIM-PF, May 13-17, 2002 Goddard Space Flight Center Formation Re-Orientation Free Space Analysis (4/4) 1 day Impulsive 1 day Continuous 1 Week Impulsive 1 Week Continuous Total  V (m/s) Acceleration (m/s 2 ) N/A1.9 e-5N/A3.81 e–7 Notes: (1) 200 km baseline, (2) 10  re-orientation of Detector SC

Final Version Flight Dynamics Page 14 MAXIM-PF, May 13-17, 2002 Goddard Space Flight Center Continuous Low Thrust Summary Detector - Phase 1 Formation Keeping 1 day 1 Formation Reorientation 1 day 2,3 Formation Reorientation (Delta) 7 days 2,3 Total  V (m/s) Acceleration (m/s 2 ) 1e-61.9 e e-7 Notes: (1) Formation Keeping Costs Highly Dependent on SRP and thus the relative A/M ratios for the spacecraft. (2) The Formation Re-Orientation Costs are based on Free Space Calculations. This number should be multiplied by a “CorrectionFactor” > 1 to account for the L2 orbit. Low Thrust Software Needed for Future Refinements. (3) The Formation Reorientation values are considered a “delta” above the baseline Formation Keeping costs. (4) All Numbers are Coarse approximations.

Final Version Flight Dynamics Page 15 MAXIM-PF, May 13-17, 2002 Goddard Space Flight Center Lissajous Orbit Option Phase 2

Final Version Flight Dynamics Page 16 MAXIM-PF, May 13-17, 2002 Goddard Space Flight Center  Possible Configuration  Optics Hub has Minimal or no Propulsion  Detector SC moves to a distance of 20,000 km from Optics Hub  FreeFlyer SC Separates from Optics hub to a maximum separation of 500 m  New Baseline May Require New Class of Continuous Thrusters for Detector SC Formation Initialization (Phase 2, km Baseline) Detector S/C (Phase 2) Optics Hub S/C 20,000 km FreeFlyer S/C 500 m 200 km Detector S/C (Phase 1)

Final Version Flight Dynamics Page 17 MAXIM-PF, May 13-17, 2002 Goddard Space Flight Center Formation Keeping (Phase 2)  Same m/s 2 from SRP Differential Assumed  Larger Baseline  Dynamics Plays a Greater Role  Control Acceleration Magnitude Depends on  Position of SC in its Orbit  Choice of Target  Sample Mission Orbit (Calculation Purposes Only)  Optics Hub at L2  Detector SC moves in a Circle about L2  20,000 km Radius  In Ecliptic Plane  Clockwise Motion (360  /yr)  Circular Restricted Three Body Problem  No Other Forces modeled  Control Acceleration  m/s 2  Combined Accel Mag  1.1 x m/s 2

Final Version Flight Dynamics Page 18 MAXIM-PF, May 13-17, 2002 Goddard Space Flight Center Formation Re-Orientation Free Space Analysis (Detector, Phase 2) 1 day Impulsive 1 day Continuous 1 Week Impulsive 1 Week Continuous Total  V (m/s) 0.8 e21.61 e21.2 e12.31 e1 Acceleration (m/s 2 ) N/A1.9 e-3N/A3.81 e-5 Notes: (1) km baseline, (2) 10  re-orientation

Final Version Flight Dynamics Page 19 MAXIM-PF, May 13-17, 2002 Goddard Space Flight Center Continuous Low Thrust Summary Detector - Phase 2 Formation Keeping 1 day 1 Formation Reorientation 1 day 2,3 Formation Reorientation (Delta) 7 days 2,3 Total  V (m/s) e22.31 e1 Acceleration (m/s 2 ) 1.1 e-51.9 e-33.8 e-5 Notes: (1) Formation Keeping Costs Highly Dependent on SRP and thus the relative A/M ratios for the spacecraft. (2) The Formation Re-Orientation Costs are based on Free Space Calculations. This number should be multiplied by a “Correction Factor” > 1 to account for the L2 orbit. Low Thrust Software Needed for Future Refinements. (3) The Formation Reorientation values are considered a “delta” above the baseline Formation Keeping costs. (4) All Numbers are Coarse approximations.

Final Version Flight Dynamics Page 20 MAXIM-PF, May 13-17, 2002 Goddard Space Flight Center Continuous Low Thrust Summary FreeFlyer - Phase 2 Formation Keeping 1 day 1 Formation Reorientation 1 day 2,3 Formation Reorientation (Delta) 7 days 2,3 Total  V (m/s) e-36 e-4 Acceleration (m/s 2 ) 1e-64.7 e-81 e-9 Notes: (1) Formation Keeping Costs Highly Dependent on SRP and thus the relative A/M ratios for the spacecraft. (2) The Formation Re-Orientation Costs are based on Free Space Calculations. This number should be multiplied by a “CorrectionFactor” > 1 to account for the L2 orbit. Low Thrust Software Needed for Future Refinements. (3) The Formation Reorientation values are considered a “delta” above the baseline Formation Keeping costs. (4) 500 m baseline (5) All Numbers are Coarse approximations.

Final Version Flight Dynamics Page 21 MAXIM-PF, May 13-17, 2002 Goddard Space Flight Center DeltaV Analysis (All Phases)

Final Version Flight Dynamics Page 22 MAXIM-PF, May 13-17, 2002 Goddard Space Flight Center DeltaV Summary (1 of 3)  L2 Propulsion Insertion Module  Carries All SC in Formation  Launch Vehicle Correction  Contingency  Mid-Course Correction (MCC)  Lissajous Orbit Insertion (LOI)  200 m/s – High Thrust

Final Version Flight Dynamics Page 23 MAXIM-PF, May 13-17, 2002 Goddard Space Flight Center DeltaV Summary (2 of 3)  Detector SC  125 m/s High Thrust for Lissajous Stabilization and Contingencies  25 m/s * 5 years  32 m/s Continuous Low Thrust for Formation Keeping in Phase 1  1e-6 m/s 2 * 1 yr  117 m/s Continuous Low Thrust for Re-Orientation (1 day) in Phase 1  (45 targets) * (1e e-5) m/s 2 * (1 day to reorient) * (Correction Factor of 1.5)  1389 m/s Continuous Low Thrust for Formation Keeping in Phase 2  1.1 e-5 m/s 2 * 4 yr  2042 m/s Continuous Low Thrust for Re-Orientation (7 day) in Phase 2  (45 targets) * (1.1 e e-5) m/s 2 * (7 day to reorient) * (Correction Factor of 1.5)

Final Version Flight Dynamics Page 24 MAXIM-PF, May 13-17, 2002 Goddard Space Flight Center DeltaV Summary ( 3 of 3)  Optics Hub  125 m/s High Thrust for Lissajous Stabilization and Contingencies  25 m/s * 5 years  FreeFlyer SC (per SC)  100 m/s High Thrust for Lissajous Stabilization and Contingencies  25 m/s * 4 years  380 m/s Continuous Low Thrust for Formation Keeping (Phase 2)  1e-6 m/s 2 * 4 yr * (Correction Factor of 3)  13 m/s Continuous Low Thrust for Re-Orientation in 1 day (Phase 2)  (45 targets) * (1 e e-8) m/s 2 * (1 day to reorient) * (Correction Factor of 3) Notes: (1) In Phase 2, the Detector SC re-orients in 1 week while the FreeFlyers re-orient in 1 day. (2) All  V values for all SC do not include engineering penalties, ACS Penalties, and cant angles. (3) Formation Re- Orientation (10  ) values include the necessary Formation Keeping contribution. (4) Double Counting of Formation Keeping costs during a Re-Orientation used to account for formation Acquisition Costs. (5) Formation Initialization Costs not explicitly listed here

Final Version Flight Dynamics Page 25 MAXIM-PF, May 13-17, 2002 Goddard Space Flight Center Flight Dynamics Technologies Required  Control Law algorithm development  Improved Control Performance  Collision Avoidance  Re-Acquisition of Formation after Re-Orientation  Simulation  Continuous Thrust model  High Fidelity Force model  Relative Navigation needed  Current Ground based Orbit Determination : 5 km position knowledge

Final Version Flight Dynamics Page 26 MAXIM-PF, May 13-17, 2002 Goddard Space Flight Center Flight Dynamics Additional Trades to Consider  Continuous Low Thrust Transfer to L2  Feasibility of Using Low Thrust for Lissajous Stabilization  Consider Surface Coatings on SC or Other Methods to minimize SRP Differentials  Formation Keeping Costs are a function of Both Position in Orbit and Choice of Target. By judicious choice of target sequence, Some  V Optimization can be Realized.  Detailed Trajectory Design Study to Include Lissajous vs. Heliocentric Trade  Heliocentric Orbits with Better Communication  Some can be Achieved Via Only Launch Vehicle Considerations  Distant Retrograde Orbits (~200 m/s)

Final Version Flight Dynamics Page 27 MAXIM-PF, May 13-17, 2002 Goddard Space Flight Center Flight Dynamics Issues and Concerns  Continuous Thrusting will make OD more Difficult  Very Difficult to Choose Class (acceleration magnitudes) of Propulsion Systems Needed  Very Coarse Estimates of Control Acceleration Magnitudes  Different Phases of Mission  New Technology: Thrusters with Greater Range of Thrust Modulation?  Relative Orbit Position Control & Knowledge Requirements Orders of Magnitude above Current Operational Capability  Collision Avoidance  Further extensive analysis required  High fidelity simulation w/ all force perturbations and sensor/actuator noise and error  Control Law Evaluation  Continuous Low Thrust Simulations  Continuous Low Thrust Trajectory Optimization Software Needed

Final Version Flight Dynamics Page 28 MAXIM-PF, May 13-17, 2002 Goddard Space Flight Center Supplementary Material

Final Version Flight Dynamics Page 29 MAXIM-PF, May 13-17, 2002 Goddard Space Flight Center Miscellany  Satellite Operators Should employ strategies to balance the fuel usage amongst all the SC in the Formation

Final Version Flight Dynamics Page 30 MAXIM-PF, May 13-17, 2002 Goddard Space Flight Center Supplementary Material – Lissajous Orbit

Final Version Flight Dynamics Page 31 MAXIM-PF, May 13-17, 2002 Goddard Space Flight Center Sample Impulsive Re-Orientation (1 of 2) (20,000 km baseline, 10  in 7 days)  Force model  Full Ephemeris  Sun/Earth/Moon/Jupiter Point Mass  SRP  Both SC Have same A/M Ratio (C r A/M = 0.013)  Initial Optics Hub State (ECI MJ2000)  UTC Gregorian Date: 23 Jan :02:45.56 UTC Julian Date:  X: km Vx: km/sec  Y: km Vy: km/sec  Z: km Vz: km/sec  Initial Detector State  Offset Position by b1 = 20000*(1, 0, 0)  Identical Velocity  Final Optics Hub State  UTC Gregorian Date: 30 Jan :02:45.56  X: e+006 km Vx: km/sec  Y: km Vy: km/sec  Z: km Vz: km/sec  Final Detector State  Offset Position by b2 = 20000*(cos(10),sin(10), 0)  Identical Velocity

Final Version Flight Dynamics Page 32 MAXIM-PF, May 13-17, 2002 Goddard Space Flight Center Sample Impulsive Re-Orientation (2 of 2)  Astrogator Simulation  First Maneuver Magnitude of 7.3 m/s  Second Maneuver Magnitude of 4.4 m/s  Total Maneuver Magnitude of 11.7 m/s  Free Space Approximations (Impulsive)  Two Equal Impulsive Maneuvers of 6 m/s  Total  V of 12 m/s  Comparison of Astrogator vs. Free Space  Fairly Good Agreement for this Sample Case  Small Re-Orientation Times  Astrogator’s Unequal Maneuver Size  Need for Previously Discussed “Correction Factor”

Final Version Flight Dynamics Page 33 MAXIM-PF, May 13-17, 2002 Goddard Space Flight Center Sample Lissajous Orbit Delta-V Budgets Direct Transfer C 3 = km 2 /kg 2 Lissajous 800 (y-amplitude ~ 800K km) Direct Transfer C 3 = km 2 /kg 2 Lissajous 400 (y-amplitude ~ 400K km) Transfer with Phasing Loops and Lunar Flyby C 3 = -2.6 km 2 /kg 2 Lissajous 200 (y-amplitude ~ 200K km) Correct Delta Inaccuracy50 m/s 20 m/s Phasing Loopsn/a 50 m/s Final Perigee Correctionn/a 15 m/s Midcourse Corrections5 m/s Lissajous Insertion2 m/s108 m/s5 m/s Lunar Shadow AvoidanceN/A10 m/s per yr Trajectory Maintenance4 m/s per yr Total, 5 years77 m/s233 m/s165 m/s Notes: (1) Total does not include engineering penalties,ACS Penalties, finite burn losses, cant angle, contingencies. Low Thrust not Considered here. (2) No Corresponding Chart for Heliocentric Orbit Option

Final Version Flight Dynamics Page 34 MAXIM-PF, May 13-17, 2002 Goddard Space Flight Center Direct vs. Phasing Loop Transfer (Lissajous Orbit Option)  Phasing Loops with Lunar Swingby  More Robust  Operationally Complex  10 Launch Days per Month (MAP 3 & 5 loop option)  Reduced C3 Costs (Not really a factor here)  Direct Transfer  Higher Risk (Little Time to React to Unforeseen Contingencies)  Simpler Operationally  22 Launch Days per Month Constellation-X Example. Courtesy Lauri Newman

Final Version Flight Dynamics Page 35 MAXIM-PF, May 13-17, 2002 Goddard Space Flight Center Libration Point Trajectory Manifolds L1L2L3 L5 L4 Y z ecliptic north pole x view from the ecliptic north pole ~1.5 x10 6 km Earth/Moon

Final Version Flight Dynamics Page 36 MAXIM-PF, May 13-17, 2002 Goddard Space Flight Center Selected Lissajous Orbit Option Issues  Define Lissajous Orbit Parameters  Phasing Loop vs. Direct Transfer  Define Maximum L2-Earth-Spacecraft Angle for Communication Purposes (MAP was 10.5 degrees)  Define how sensitive Spacecraft is to Shadow in Phasing Loops  Review Lessons Learned from Other Libration Point Missions such as MAP & Triana  Insure that Thrusters are sized large enough to produce Desired DeltaV in a Reasonable time (For Transfer Trajectory)

Final Version Flight Dynamics Page 37 MAXIM-PF, May 13-17, 2002 Goddard Space Flight Center Large Lissajous / Direct Transfer projection onto ecliptic plane (ie, top view) projection onto xz plane (ie, side view) projection onto yz plane (ie, view from earth)

Final Version Flight Dynamics Page 38 MAXIM-PF, May 13-17, 2002 Goddard Space Flight Center Small Lissajous / Direct Transfer projection onto yz plane (ie, view from earth) projection onto ecliptic plane (ie, top view) projection onto xz plane (ie, side view)

Final Version Flight Dynamics Page 39 MAXIM-PF, May 13-17, 2002 Goddard Space Flight Center Small Lissajous / Lunar Gravity Assist Y-Amp ~ 200k Z-Amp ~ 300k projection onto yz plane (ie, view from earth) projection onto ecliptic plane (ie, top view) projection onto xz plane (ie, side view)

Final Version Flight Dynamics Page 40 MAXIM-PF, May 13-17, 2002 Goddard Space Flight Center Triana (L1 Lissajous Orbit) DSN/USN Support Requirements (Example from Triana Peer Review) Note: Since USN had planned Dedicated Triana Support, Some of these Requirements may be Overkill. Data Courtesy Greg Marr.

Final Version Flight Dynamics Page 41 MAXIM-PF, May 13-17, 2002 Goddard Space Flight Center MAP Lunar Shadows (L2 Lissajous Mission Orbit)  Sample Worst Cases  MAP is a small amplitude Lissajous  Moon Farther from L2  8 Hour Shadow with Maximum Depth of 4.5%  Moon Closer to L2  6 Hour Shadow with Maximum Depth of 13% Note: Data courtesy Mike Mesarch

Final Version Flight Dynamics Page 42 MAXIM-PF, May 13-17, 2002 Goddard Space Flight Center Supplementary Material – Heliocentric Orbit

Final Version Flight Dynamics Page 43 MAXIM-PF, May 13-17, 2002 Goddard Space Flight Center MAXIM-PF Range From Earth (Heliocentric Orbit Option) Reference: August 99 MAXIM IMDC Study

Final Version Flight Dynamics Page 44 MAXIM-PF, May 13-17, 2002 Goddard Space Flight Center MAXIM-PF Trajectory in Solar Rotating Coordinates (Heliocentric Orbit Option) Reference: August 99 MAXIM IMDC Study

Final Version Flight Dynamics Page 45 MAXIM-PF, May 13-17, 2002 Goddard Space Flight Center Heliocentric Orbit Option Phase 1

Final Version Flight Dynamics Page 46 MAXIM-PF, May 13-17, 2002 Goddard Space Flight Center Formation Initialization  One LV with a C3 of 0.4 km 2 /s 2  Needed to put the trajectories beyond Earth’s sphere of influence (SOI is ~10 6 km) Relatively Quickly  One SC is maneuvered to the shifted orbit 200 km away from the other’s origin

Final Version Flight Dynamics Page 47 MAXIM-PF, May 13-17, 2002 Goddard Space Flight Center  Desired Characteristics  Two S/C in formation, 200 km apart  Maintain inertial orientation of SC-to-SC line for 1 week observation  One SC follows a circular, heliocentric orbit  Other SC follows a shifted, circular, heliocentric trajectory with orbit plane parallel to the plane of the first SC  Center of shifted trajectory lies on the Sun-target line 200 km from Sun  Driving Requirements  Time allocated for reorienting the SC-to-SC line  SC-to-SC line remains inertially fixed during observation Heliocentric Orbit Description (Phase 1)

Final Version Flight Dynamics Page 48 MAXIM-PF, May 13-17, 2002 Goddard Space Flight Center Formation Keeping (1/2) (Heliocentric Orbit, Phase 1, 200 km Baseline)  Apply control accelerations continuously to maintain the inertial orientation of the SC-to-SC line  ~0.01 m/s per week  Only Solar Gravity modeled  Circular Earth Orbit about Sun  SRP Differential Acceleration not considered here (Very Important Term)  Maximum control accelerations are needed when the trajectories are coplanar (it’s counter-intuitive)  0.8 x to 1.6 x m/s 2  8 to 16 micro-newton thrust for a 1000 kg SC Control acceleration magnitude -vs- time since station-keeping starts

Final Version Flight Dynamics Page 49 MAXIM-PF, May 13-17, 2002 Goddard Space Flight Center Formation Keeping (2/2) (Heliocentric Orbit, Phase 1, 200 km Baseline)  Control Acceleration Magnitude Depends on  Position of SC in its Orbit  Choice of Target  Control Acceleration Magnitude Varies (Approximately) Linearly with Baseline Assuming:  For Our Range of Baselines  Ecliptic Target with RA=DEC=0  Only Solar Gravity modeled  Circular Orbit

Final Version Flight Dynamics Page 50 MAXIM-PF, May 13-17, 2002 Goddard Space Flight Center Supplementary Material – General

Final Version Flight Dynamics Page 51 MAXIM-PF, May 13-17, 2002 Goddard Space Flight Center Formation Re-Orientation Free Space Analysis Revisited  Impulsive Analysis  One Burn at Observation 1 (Magnitude V1) & One Burn (Same Magnitude, Opposite Direction) at Observation 2  V1 (m/s) = Distance (m) / t 0 (s)  Total  V (m/s) = 2 V1 = 2 * Distance (m) / t 0 (s) (t 0 is time to re-orient)  Continuous Thrust Analysis  Acceleration is a positive constant (magnitude A) from t = 0 to t = t 0 /2  Acceleration is a negative constant (same magnitude) from time t 0 /2 to time, t 0  At time, t=0 & t = t 0, Velocity is 0  At time, t= t 0 /2, Velocity reaches a maximum of V2 = 2 V1 = 2 * Distance /t 0  Total  V (m/s) is Twice that of Impulsive Case: 4 * Distance / t 0  A = Distance / (t 0 /2) 2 = 2 V2 / t 0 = 4 V1 / t 0

Final Version Flight Dynamics Page 52 MAXIM-PF, May 13-17, 2002 Goddard Space Flight Center References (1 of 3)  Marr, Cooley, Franz, Roberts, Triana Trajectory Design Peer Review,  Cuevas, Newman, Mesarch, Woodard, An Overview of Trajectory Design Operations for the MAP Mission, AIAA , AIAA Astrodynamics Specialist Conference, August  Mesarch, Andrews, The Maneuver Planning Process for the MAP Mission, AIAA , AIAA Astrodynamics Specialist Conference, August  Mesarch, Contingency Planning for the MAP Mission, AIAA , AIAA Astrodynamics Specialist Conference, August  L. Newman, Constellation-X Reference Mission Description Document, Govind Gadwal, ed.,  Mesarch, Vaughn, Concha, Flight Dynamics IMDC Study for the MAXIM Mission, August 1999.

Final Version Flight Dynamics Page 53 MAXIM-PF, May 13-17, 2002 Goddard Space Flight Center References (2 of 3)  Cooley, Marr, Starin, Petruzzo, Flight Dynamics IMDC Study for the Fresnel Lens Gamma Ray Telescope, January  Cooley, Marr, Starin, Petruzzo, Flight Dynamics IMDC Study for the Fresnel Lens Gamma Ray PathFinderTelescope, January  Concha, Cooley, Folta, Hamilton, Flight Dynamics IMDC Study for the Stellar Imager, July  Markley, Maxim Mission White Paper, January 31,  Grady, MAXIM Pathfinder Mission Concept Design Powerpoint Presentation, MPF Mission Definition Team Meeting, September 18,  Wertz, ed., Spacecraft Mission Analysis and Design, 3 rd Edition, Microcosm, 1999.

Final Version Flight Dynamics Page 54 MAXIM-PF, May 13-17, 2002 Goddard Space Flight Center References (3 of 3)  Luquette, Sanner, A nonlinear approach to spacecraft formation control in the vicinity of a collinear libration point, AAS , 2001.