Trajectory Design For A Visible Geosynchronous Earth Imager Edmund M. C. Kong SSL Graduate Research Assistant Prof David W. Miller Director, MIT Space Systems Lab Dr. Raymond J. Sedwick Postdoctoral Associate, MIT Space Systems Lab AIAA Space Technology Conference & Exposition Albuquerque, New Mexico 30 September, 1999 Space Systems Laboratory Massachusetts Institute of Technology Introduction Objective : To compare the different imaging configurations for a Separated Spacecraft Interferometer operating from an Earth’s orbit Outline : – Interferometric requirements & Orbit Selection – Equations of Motions (Hill’s Equations) – Steered Planar Array – Propellant Free Array: Collector S/C – Results – Summary Space Systems Laboratory Massachusetts Institute of Technology Interferometric Requirements & Orbit Selection Interferometric Requirements: Reqt 1. Equal science light pathlength for visible imaging Reqt 2. Axi-symmetric angular resolution about LOS Far-field assumption ? Array sees planar wavefronts from targets y x z Orbit Selection: Geosynchronous ? Higher altitude, lower perturbative effects (eg. J 2 ) Space Systems Laboratory Massachusetts Institute of Technology Equations of Motions Assumption : First order perturbation about natural circular Keplerian orbit ( c r o s s - r a n g e ) z x N y ( ze n i t h - n a d i r ) ( v e l o c i t y v e c t o r ) S znza xnya ynxnxa z y x 2 2 2 23 += += ??= && &&& &&& dtaaa life T zyx ∫ ++=? 0 222 V Hill’s Equations : Total Spacecraft Velocity Increment : Example : ?V required to hold a spacecraft stationary at (x,y,z) 222 9nV zxT life +=? ?V required :Spacecraft instantaneous acceleration : 0= y a zna z 2 =xna x 2 3?= Space Systems Laboratory Massachusetts Institute of Technology DSS Architecture 1 Constraint collector spacecraft to a local horizontal circular trajectory with combiner spacecraft at the center (Reqts. 1 & 2) 55.1/V 2 =? lifeo TRn () ()? ? ? ? ? ? ? ? ? ? α+ α+±= ? ? ? ? ? ? ? ? ? ? ′ ′ ′ ntR ntR z y x o o cos sin 0x, a z y L O S ψ φ x' b c, z' y ' ?No ?V for stationary combiner spacecraft at (0,y,0) ? ?V for collector spacecraft ?V Requirement Average collector s/c ?V at GEO altitude : ? ? ? ? ? ? ? ? ? ? ′ ′ ′ ? ? ? ? ? ? ? ? ? ? ψψ ψ?ψ ? ? ? ? ? ? ? ? ? ? φφ φ?φ= ? ? ? ? ? ? ? ? ? ? z y x z y x Hill 100 0cossin 0sincos sinsin0 sincos0 001 Space Systems Laboratory Massachusetts Institute of Technology DSS Architecture 2 -1.5 0 1.5 -1.5 0 1.5 -1.5 0 1.5 Velocity Vector (y/R o ) R z = R o Cross Axis (z/R o ) Z e n i th (x /R o ) -180 -90 0 90 180 0 30 60 90 120 150 180 (R z = R o ) R z = 0 (R z = ∞) R z = R o (R z = 0) [R z = -0.87R o ] R z = ∞ (R z = -∞) [R z = 0.87R o ] (R z = -R o ) R z = -R o R z = -∞ LEO GEO ψ ( degr ees ) φ (degrees) ()( ) ? ? ? ? ? ? ? ? ? ? ± ? = ? ? ? ? ? ? ? ? ? ? 4 0 16316 22 p ppR z y x o Focus () ? ? ? ? ? ? ? ? ? ? ± = ? ? ? ? ? ? ? ? ? ? ntR ntR ntR z y x z o o Collector cos sin cos2 m Constraint the projection of the collector spacecraft’s trajectory to circular (Reqt. 2) ? Propellant free trajectories - (Project 2 x 1 ellipse in velocity plane) Intersection between a plane and a circular paraboloid results in an ellipse ? Placed combiner spacecraft placed at focus for equal pathlength (Reqt. 1) ?for R z = R o Vary R z : (-∞,∞) Space Systems Laboratory Massachusetts Institute of Technology 0 0.5 1 1.5 2 2.5 3 3.5 4 0 1 2 3 4 5 6 7 8 9 10 p/R o ? V/ n 2 R o T lif e (2.2076,0.5642) Optimum focus : o Rp 2076.2= lifeo TRn 2 5642.0V =? DSS Architecture 2 (cont.) A family of paraboloids can fit onto the free elliptical trajectories ? Locus of foci maps out a hyperbola ?for R z = R o -1.5 -1 -0.5 0 0.5 1 1.5 -1.5 -1 -0.5 0 0.5 1 1.5 -1 -0.5 0 0.5 1 y/R o (velocity vector) Circular Paraboloid Ellipse ← Optimal Focus (p/R o =2.2076) Projected Circle z/R o (Cross axis) Hyperbola (Foci) x/ R o ( Z en i t h N a di r ) z zR x o 4 3 22 ± ? = ?V requirement: ?No ?V required for collector spacecraft ? Only need ?V to hold combiner spacecraft at paraboloid’s focus Space Systems Laboratory Massachusetts Institute of Technology Optical Delay Lines -1.2 -0.9 -0.6 -0.3 0 0.3 0.6 0.9 1.2 -0.6 -0.3 0 0.3 0.6 x 1 " x 2 " Imaginary Paraboloid Elliptical Trajectory D 1 D 2 Delay/R o = |D 1 - D 2 | φ = 0 o , ψ = 162 o Combiner (Focus) d target Cross Range (z/R o ) Z e n i th ( x / R o ) 180900?90?180 180 90 0 GEO LEO φ (degrees) ψ (degrees) Delay/R o 0 1 2 3 4 Steering with optical delay lines ? Collector s/c follow R z = R o elliptical trajectory from Architecture 2 ? Delay lines to image off-nadir targets (Reqt. 1) Hill z y x z y x ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ? ? ? ? ? ?= ? ? ? ? ? ? ? ? ? ? ′ ′ ′ φφ φφψψ ψψ sinsin0 sincos0 001 100 0cossin 0sincos 2 2 2 1 256 25 8 5 cos 16 51 4 )(cos n nn n o P PntP P nt R D +++ ? ? ? ? ? ? ? ? ++= At GEO ? Maximum delay length from GEO (x’,D) = 0.310R o ? Minimum semi-minor axis projection (y’,z’) = 0.914R o Collector s/c trajectory in propagation vector’s (x’) frame: Collector-Combiner s/c distance: Space Systems Laboratory Massachusetts Institute of Technology Mission Parameters Components Steered Planar ODL Combiner S/C 182.1 kg 182.1 kg Combiner Propellant - ?V/(n 2 R o T life ) = 0.56 Collector S/C 87.1 kg 87.1 kg Collector Delay Lines - 0.34R o Collector Propellant ?V/(n 2 R o T life ) = 1.55 - Spacecraft Mass estimates from initial Deep Space 3 (DS3) design ?T life = 5 years ?R o = 500 m (DS3 - 1000 m baseline) For each spacecraft ? Determine ?V ? Propellant mass from Rocket equation 1 V exp ? ? ? ? ? ? ? ? ? ? = gIm m spd p m p - Propellant Mass (kg) m d - Spacecraft Dry Mass (kg) I sp - Specific impulse (sec) Place ODL on Collector S/C ? Ease of operation ? Lower overall dry mass and therefore, lower system mass g - Earth’s gravity (9.81 m/sec) Space Systems Laboratory Massachusetts Institute of Technology Impact of ODL General Observations ? Relatively insensitive to the number of collector s/c (> 4 collector) ? Trading between propellant and ODL mass R o = 500 m ? Break even point I sp = 250 s (DLC = 0. 1 kg/m) ? Arch 1 : m comb = 182.1, m coll = 114.1 ? Arch 2 : m comb = 200.4, m coll = 104.1 R o = 50 m ? Break even point I sp = 220 s (DLC = 0.1 kg/m) ? Arch 1 : m comb = 182.1, m coll = 89.7 ? Arch 2 : m comb = 184.1, m coll = 88.8 100 200 300 400 500 600 700 800 900 1000 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Specific Impulse (secs) D e l a y Lengt h C onv er s i on ( k g/ m ) R o = 500 m R o = 50 m (exp( / ) 1) 0.34 coll sp o mVIg DLC R ? ? ≈ Space Systems Laboratory Massachusetts Institute of Technology Summary (1) ? Interferometric Requirements y x z ? Equations of Motions – Hill’s Equations – ?V Calculation ? DSS Architecture 1 – ?V for collector spacecraft only Space Systems Laboratory Massachusetts Institute of Technology Summary (2) ? DSS Architecture 2 – Free ?V trajectories for collector spacecraft – Minimum ?V combiner spacecraft location – Exploitation of conic sections ? Results – Delay Length vs Specific Impulse cross over point ? Optical Delay Lines – Delay lines to steer array’s LOS 100 200 300 400 500 600 700 800 900 1000 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Specific Impulse (secs) D e l a y L engt h C onv er s i on ( k g/ m ) R o = 500 m R o = 50 m -1.2 -0.9 -0.6 -0.3 0 0.3 0.6 0.9 1.2 -0.6 -0.3 0 0.3 0.6 x 1 " x 2 " Imaginary Paraboloid Elliptical Trajectory D 1 D 2 Delay/R o = |D 1 - D 2 | φ = 0 o , ψ = 162 o Combiner (Focus) d target Cross Range (z/R o ) Z eni t h (x / R o ) -180 -90 0 90 180 0 30 60 90 120 150 180 (R z = R o ) R z = 0 (R z = ∞) R z = R o (R z = 0) [R z = -0.87R o ] R z = ∞ (R z = -∞) [R z = 0.87R o ] (R z = -R o ) R z = -R o R z = -∞ LEO GEO ψ ( deg r e es ) φ (degrees) -1.5 -1 -0.5 0 0.5 1 1.5 -1.5 -1 -0.5 0 0.5 1 1.5 -1 -0.5 0 0.5 1 y/R o (velocity vector) Circular Paraboloid Ellipse ← Optimal Focus (p/R o =2.2076) Projected Circle z/R o (Cross axis) Hyperbola (Foci) x/ R o (Z e n i t h Na d i r )