# Presentation: 513108

## Micro-Arcsecond X-ray Imaging Mission, Pathfinder (MAXIM-PF)

**Rich Luquette**
**Greg Marr**
**Scott Starin**

**Flight Dynamics**

**May 13-17, 2002**

**Micro-Arcsecond X-ray Imaging Mission, Pathfinder (MAXIM-PF)**

## Requirements & Assumptions (1 of 2) 

** **Phase 1

** **Phase 2

- 5cm control
- 15  m Knowledge

- Optics Hub S/C

- Detector S/C

- 20,000 km +/- 5 m

**FreeFlyer S/C**

- 100-500 m separation
- Control to ~10 microns

- Detector S/C

- Optics Hub S/C

## 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

- 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) **

## 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)**

## 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**

## Lissajous Orbit Option 
Phase 1

**Phase 1**

## 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

- 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)**

## 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

## Formation KeepingSolar Radiation Pressure (SRP)

**Acceleration Magnitude (1 AU) ******** 4.5 x 10****-6 ****(1+r) A/M  (m/s****2****)**
- A = Cross Sectional Area exposed to Sun (m2)
- M = Mass of Spacecraft (kg)
- r = Reflection Factor.  (r  [0,1])
- Approximate Result for all Mission Orbits Considered
- SMAD (3rd Edition, not 2nd edition)
**SRP Acceleration Magnitude Differential Between 2 Spacecraft**
-  4.5 x 10-6 | (1+r1) (A1/m1) – (1+r2) (A2/m2)|
- Assumed Dominant Term for 200 km Baseline (CRTBP model)
**Assume Control Acceleration Magnitude ******** 10****-6 ****m/s****2**** Needed**

## 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)**

## 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

## 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

## Formation Re-OrientationFree Space Analysis (4/4) 

|  | 1 day
Impulsive | 1 day
Continuous | 1 Week
Impulsive | 1 Week
Continuous |
| --- | --- | --- | --- | --- |
| Total V (m/s) | 0.8 | 1.61 | 0.12 | 0.23 |
| Acceleration (m/s2) | N/A | 1.9 e-5 | N/A | 3.81 e–7 |

- Notes:  (1)  200 km baseline, (2) 10 re-orientation of Detector SC

## Continuous Low Thrust SummaryDetector - Phase 1

|  | Formation Keeping
1 day1 | Formation Reorientation
1 day2,3 | Formation Reorientation (Delta)
7 days2,3 |
| --- | --- | --- | --- |
| Total V (m/s) | 0.0864 | 1.61 | 0.23 |
| Acceleration (m/s2) | 1e-6 | 1.9 e-5 | 3.81 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.

## Lissajous Orbit Option 

Phase 2

**Phase 2**

## 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

- 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, 20000 km Baseline)**

## Formation Keeping(Phase 2)

**Same 10****-6 ****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  10-5 m/s2
**Combined Accel Mag ******** 1.1 x 10****-5 ****m/s****2**

## 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 e2 | 1.61 e2 | 1.2 e1 | 2.31 e1 |
| Acceleration (m/s2) | N/A | 1.9 e-3 | N/A | 3.81 e-5 |

- Notes:  (1)  20000 km baseline, (2) 10 re-orientation

## Continuous Low Thrust SummaryDetector - Phase 2 

|  | Formation Keeping
1 day1 | Formation Reorientation
1 day2,3 | Formation Reorientation (Delta)
7 days2,3 |
| --- | --- | --- | --- |
| Total V (m/s) | 0.95 | 1.61 e2 | 2.31 e1 |
| Acceleration (m/s2) | 1.1 e-5 | 1.9 e-3 | 3.8 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.

## Continuous Low Thrust SummaryFreeFlyer - Phase 2 

|  | Formation Keeping
1 day1 | Formation Reorientation 
1 day2,3 | Formation Reorientation (Delta)
7 days2,3 |
| --- | --- | --- | --- |
| Total V (m/s) | 0.0864 | 4.1 e-3 | 6 e-4 |
| Acceleration (m/s2) | 1e-6 | 4.7 e-8 | 1 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.

## DeltaV Analysis (All Phases)

## 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

## 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/s2 * 1 yr
- 117 m/s Continuous Low Thrust for Re-Orientation (1 day) in Phase 1
- (45 targets) * (1e-6 + 1.9 e-5) m/s2 * (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/s2 * 4 yr
- 2042 m/s Continuous Low Thrust for Re-Orientation (7 day) in Phase 2
- (45 targets) * (1.1 e-5 + 3.8 e-5) m/s2 * (7 day to reorient) * (Correction Factor of 1.5)

## 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/s2 * 4 yr * (Correction Factor of 3)
- 13 m/s Continuous Low Thrust for Re-Orientation in 1 day (Phase 2)
- (45 targets) * (1 e-6 + 4.7 e-8) m/s2 * (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

## 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

## 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)**

## 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

## Supplementary
Material

**Material**

## Miscellany

**Satellite Operators Should employ strategies to balance the fuel usage amongst all the SC in the Formation**

## Supplementary
Material – Lissajous Orbit

**Material – Lissajous Orbit**

## 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 (Cr A/M = 0.013)
**Initial Optics Hub State (ECI MJ2000)**
**UTC Gregorian Date: 23 Jan 2003 05:02:45.56  UTC Julian Date: 2452662.71024955                                                                                                                **
**         ****X:  -993733.7803065266900000 km              Vx:       -0.3149752411661734 km/sec     **
**         ****Y:   913746.5347422765300000 km              Vy:       -0.2540742769815505 km/sec     **
**         ****Z:   396534.8804631549300000 km              Vz:       -0.0421253073023613 km/sec    **
**Initial Detector State**
- Offset Position by b1 = 20000*(1, 0, 0)
- Identical Velocity
**Final Optics Hub State**
**UTC Gregorian Date: 30 Jan 2003 05:02:45.56                                                                                               **
**         ****X:  -1.1621310681357966e+006 km              Vx:       -0.2462221518003097 km/sec     **
**         ****Y:   757799.1103539797500000 km              Vy:       -0.2561905904275567 km/sec     **
**         ****Z:   369320.0722157274700000 km              Vz:       -0.0457073285913774 km/sec**
** ****Final Detector State**
- Offset Position by b2 = 20000*(cos(10),sin(10), 0)
- Identical Velocity

## 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”

## Sample Lissajous Orbit Delta-V Budgets

|  | Direct Transfer
C3 = -0.677 km2/kg2
Lissajous 800
(y-amplitude ~ 800K km) | Direct Transfer
C3 = -0.677 km2/kg2
Lissajous  400
(y-amplitude ~ 400K km) | Transfer with Phasing Loops and
Lunar Flyby
C3 = -2.6 km2/kg2
Lissajous 200
(y-amplitude ~ 200K km) |
| --- | --- | --- | --- |
| Correct Delta Inaccuracy | 50 m/s | 50 m/s | 20 m/s |
| Phasing Loops | n/a | n/a | 50 m/s |
| Final Perigee Correction | n/a | n/a | 15  m/s |
| Midcourse Corrections | 5 m/s | 5 m/s | 5 m/s |
| Lissajous Insertion | 2 m/s | 108 m/s | 5 m/s |
| Lunar Shadow Avoidance | N/A | 10 m/s per yr | 10 m/s per yr |
| Trajectory Maintenance | 4 m/s per yr | 4 m/s per yr | 4 m/s per yr |
| Total, 5 years | 77 m/s | 233 m/s | 165 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**

## 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

## Libration Point Trajectory Manifolds

- L1

- L2

- L3

- L5

- L4

- Y

**z**
**ecliptic**
**north**
**pole**

- x

**view from the**
**ecliptic north pole**

- ~1.5 x106  km

**Earth/Moon**

## 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)

## 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)

## 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)

## 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)

## 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.

## 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

## Supplementary
Material – Heliocentric Orbit

**Material – Heliocentric Orbit**

## MAXIM-PF Range From Earth(Heliocentric Orbit Option)

- Reference:   August 99 MAXIM IMDC Study

## MAXIM-PF Trajectory in Solar Rotating Coordinates(Heliocentric Orbit Option)

- Reference:   August 99 MAXIM IMDC Study

## Heliocentric Orbit Option 

Phase 1

**Phase 1**

## 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 ~106 km) Relatively Quickly
**One SC is maneuvered to the shifted orbit 200 km away from the other’s origin**

## 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

- 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)**

## 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 10-8  to  1.6 x 10-8 m/s2
- 8 to 16 micro-newton thrust
-    for a 1000 kg SC

**Control acceleration magnitude**
**-vs-**
**time since station-keeping starts**

## 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

## Supplementary
Material – General

**Material – General**

## 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) / t0 (s)
- Total V (m/s) = 2 V1 =  2 * Distance (m) / t0(s)         (t0 is time to re-orient)
**Continuous Thrust Analysis**
- Acceleration is a positive constant (magnitude A) from t = 0 to t = t0/2
- Acceleration is a negative constant (same magnitude) from time t0 /2 to time, t0
- At time, t=0 & t = t0, Velocity is 0
- At time, t= t0/2, Velocity reaches a maximum of V2 = 2 V1 = 2 * Distance /t0
- Total V (m/s) is Twice that of Impulsive Case:   4 * Distance  / t0
- A = Distance / (t0/2)2   =   2 V2 / t0 = 4 V1 / t0

## References (1 of 3)

- Marr, Cooley, Franz, Roberts, Triana Trajectory Design Peer Review, 2001.
- Cuevas, Newman, Mesarch, Woodard, An Overview of Trajectory Design Operations for the MAP Mission, AIAA 2002-4425, AIAA Astrodynamics Specialist Conference, August 2002.
- Mesarch, Andrews, The Maneuver Planning Process for the MAP Mission, AIAA 2002-4427, AIAA Astrodynamics Specialist Conference, August 2002.
-  Mesarch, Contingency Planning for the MAP Mission, AIAA 2002-4426, AIAA Astrodynamics Specialist Conference, August 2002.
- L. Newman, Constellation-X Reference Mission Description Document, Govind Gadwal, ed., 2002.
- Mesarch, Vaughn, Concha, Flight Dynamics IMDC Study for the MAXIM Mission, August 1999.

## References (2 of 3)

- Cooley, Marr, Starin, Petruzzo, Flight Dynamics IMDC Study for the Fresnel Lens Gamma Ray Telescope, January 2002.
- Cooley, Marr, Starin, Petruzzo, Flight Dynamics IMDC Study for the Fresnel Lens Gamma Ray PathFinderTelescope, January 2002.
- Concha, Cooley, Folta, Hamilton, Flight Dynamics IMDC Study for the Stellar Imager, July 2001.
- Markley, Maxim Mission White Paper, January 31, 2002.
- Grady, MAXIM Pathfinder Mission Concept Design Powerpoint Presentation, MPF Mission Definition Team Meeting, September 18, 2000.
- Wertz, ed., Spacecraft Mission Analysis and Design, 3rd Edition, Microcosm, 1999.

## References (3 of 3)

**Luquette, Sanner, A nonlinear approach to spacecraft formation control in the vicinity of a collinear libration point, AAS001-330,  2001.   **