麻省理工学院：《Satellite Engineering》Lecture 13 Trajectory Design For A Visible

Trajectory Design For A visibl Geosynchronous Earth Imager Edmund m.c. Kor SSL Graduate Research Assistant Prof david w. miller ctor, 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

Trajectory Design For A Visible Trajectory Design For A Visible Geosynchronous Earth Imager 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 & Expositi o n Albuquerque, New Mexico 30 September, 1999

Introduction Objective: To compare the different imaging configurations for a Separated Spacecraft Interferometer operating from an Earths 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

Space Systems Laboratory Massachusetts Institute of Technology Introduction Introduction Objective : To compare the different imaging configurations for a To compare the different imaging configurations for a Separated Spacecraft Interferometer operating from an Separated Spacecraft Interferometer operating from an Earth’s orbit 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

Interferometric Requirements Orbit Selection /lo Interferometric Requirements Reqt 1. Equal science light pathlength for visible imaging Regt 2. Axi-symmetric angular resolution about LOS Far-field assumption Array sees planar wavefronts from targets Orbit Selection Geosynchronous Higher altitude, lower perturbative effects(eg J2) Space Systems Laboratory Massachusetts Institute of Technology

Space Systems Laboratory Massachusetts Institute of Technology Interferometric Interferometric Requirements & Orbit Selection 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. J2)

Equations of Motions assumption First order perturbation about natural circular Keplerian orbit Hills Equations x-3n'x-2ny = y+2nx n Total Spacecraft Velocity Increment △V m212, atat a Example: AV required to hold a spacecraft stationary at (x, y, z) Spacecraft instantaneous acceleration △ V required n X a.=n2 △V=nTmv9x2+z Space Systems Laboratory Massachusetts Institute of Technology

Space Systems Laboratory Massachusetts Institute of Technology Equations of Motions Equations of Motions Assumption : First order perturbation about natural circular Keplerian orbit (cross-range) z x N y (zenith-nadir) (velocity vector) S a z n z a y nx a x n x ny z y x 2 2 2 3 2 = + = + = − − && && & && & a a a dt Tlife ∆ = ∫ x + y + z 0 2 2 2 V Hill’s Equations : Total Spacecraft Velocity Increment : Example : ∆V required to hold a spacecraft stationary at (x,y,z) 2 2 2 V n T 9x z ∆ = life + Spacecraft instantaneous acceleration : ∆V required : ay = 0 a n z z 2 ax n x = 2 = −3

DSS Architecture 1 Constraint collector spacecraft to a local horizontal circular trajectory with combiner spacecraft at the center(Regts. 1& 2) △ V Requirement No AV for stationary combiner spacecraft at(o,y, 0 y|=±Rsin(m+a) AV for collector spacecraft R, cos(nt+a) 100 0 y -siny 0x s-sinφ sin cos y0y sin o sin⊥0 y LEO Average collector s/C AV at geo altitude △V/n2RT=1.55 Collector AV/n".Tif Space Systems Laboratory Massachusetts Institute of Technology

Space Systems Laboratory Massachusetts Institute of Technology DSS Architecture 1 DSS Architecture 1 Constraint collector spacecraft to a local horizontal circular trajectory with combiner spacecraft at the center (Reqts. 1 & 2) V / 1.55 2 ∆ n RoTlife = ( ) ( ) ⎥⎥⎥⎦⎤ ⎢⎢⎢⎣⎡ + α = ± + α ⎥⎥⎥⎦⎤ ⎢⎢⎢⎣⎡ ′′′ R nt R nt zyx o ocossin x, a 0 z y LOS 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 : ⎥⎥⎥⎦⎤ ⎢⎢⎢⎣⎡ ′′′ ⎥⎥⎥⎦⎤ ⎢⎢⎢⎣⎡ ψ ψ ψ − ψ ⎥⎥⎥⎦⎤ ⎢⎢⎢⎣⎡ φ φ = φ − φ ⎥⎥⎥⎦⎤ ⎢⎢⎢⎣⎡ zyx zyx Hill 0 0 1 sin cos 0 cos sin 0 0 sin sin 0 cos sin 1 0 0

DSS Architecture 2 Constraint the projection of the collector Vary Rz: (-o0, oo) spacecrafts trajectory to circular(Regt. 2) Propellant free trajectories-(Project 2 X 1 ellipse in velocity plane) (R/2) cos nt +R Sin nt R cosnt Velocity Vector (Ro Intersection between a plane and a circular 2=R paraboloid results in an ellipse R=08N Placed combiner spacecraft placed at focus for equal pathlength( Reqt. 1) forr_= r 6R2-3p2)(6p) Fp/(16 LEO =.8TR Focus ±p/4 GEO ◆( degrees) Space Systems Laboratory Massachusetts Institute of Technology

Space Systems Laboratory Massachusetts Institute of Technology DSS Architecture 2 DSS Architecture 2 -1.5 0 1.5 -1.5 0 1.5 -1.5 0 1.5 Velocity Vector (y/Ro) R z = R o Cross Axis (z/Ro) Z e n ith (x/Ro ) -180 -90 0 90 180 0 30 60 90 120 150 180 (Rz = Ro) R z = 0 (Rz = ∞) R z = R o (Rz = 0) [Rz = -0.87Ro] R z = ∞ (Rz = -∞) [Rz = 0.87Ro] (Rz = -Ro) R z = -R o R z = -∞ LEO GEO ψ (degrees ) φ (degrees) ( ) ( ) ⎥⎥⎥⎦⎤ ⎢⎢⎢⎣⎡ ±− = ⎥⎥⎥⎦⎤ ⎢⎢⎢⎣⎡ 4 0 16 3 16 2 2 p R p p zyx o Focus ( ) ⎥⎥⎥⎦⎤ ⎢⎢⎢⎣⎡± = ⎥⎥⎥⎦⎤ ⎢⎢⎢⎣⎡ R nt R nt R nt zyx z o o Collector cossin 2 cos 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 Rz = Ro Vary Rz : (-∞,∞)

DSS Architecture 2(cont,) A family of paraboloids can fit onto the free elliptical trajectories Locus of foci maps out a hyperbola for R,=Ro R2-3 ±4z Circular Paraboloid 0. Optimum focus -0.5 P 2.2076R △V=0.5642nRT -05 15151050051-15 △ V requirement Z/R(Cross axis) y/R ( velocity vector) No AV required for collector spacecraft Only need Av to hold combiner spacecraft at paraboloids focus Space Systems Laboratory Massachusetts Institute of Technology

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/Ro ∆V/ n 2 R o Tlif e (2.2076,0.5642) Optimum focus : Ro p = 2.2076 RoTlife n2 ∆V = 0.5642 DSS Architecture 2 (cont.) DSS Architecture 2 (cont.) A family of paraboloids can fit onto the free elliptical trajectories • Locus of foci maps out a hyperbola • for Rz = Ro -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/Ro(velocity vector) Circular Paraboloid Ellipse ← Optimal Focus (p/Ro=2.2076) Projected Circle z/R o (Cross axis) Hyperbola (Foci) x/ R o ( Zen i t h N adi r ) z R z x o 4 3 2 2 ± − = ∆V requirement: • No ∆V required for collector spacecraft • Only need ∆V to hold combiner spacecraft at paraboloid’s focus

Optical Delay lines Steering with optical delay lines Collector s/c trajectory in propagation Collector s/c follow R, =Ro elliptical trajectory vectors(X)frame from architecture 2 cosy siny 01 0 Delay lines to image off-nadir targets(Regt. 1) =I-siny cosy 00 cos p sin p Collector-Combiner s/c distance 010- sing sing=」mn D=R (osm)2(1 v 4Ip+iPn cosnt +8+ 25-PA+ P2 Imaginary Paraboloid Elliptical Trajectory 下 At Geo Maximum delay length from GEO (X,D) Delav/R D =0.310R 06◆=0,v=162° -03 Minimum semi-minor axis projection 0.6 Cross Range(Z/Ro) yz)=0.914R。 Space Systems Laboratory Massachusetts Institute of Technology

Space Systems Laboratory Massachusetts Institute of Technology Optical Delay Lines 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 x1" x2" Imaginary Paraboloid Elliptical Trajectory D1 D2 Delay/Ro = |D1 - D2| φ = 0o, ψ = 162o Combiner (Focus) dtarget Cross Range (z/Ro) Ze n ith ( x / R o ) −180 −90 0 90 180 180 90 0 GEO LEO φ (degrees) ψ (degrees) Delay/Ro 0 1 2 3 4 Steering with optical delay lines • Collector s/c follow Rz = Ro elliptical trajectory from Architecture 2 • Delay lines to image off-nadir targets (Reqt. 1) Hill z y x z y x ⎥⎥⎥⎦⎤ ⎢⎢⎢⎣⎡⎥⎥⎥⎦⎤ ⎢⎢⎢⎣⎡⎥ − ⎥⎥⎦⎤ ⎢⎢⎢⎣⎡ = − ⎥⎥⎥⎦⎤ ⎢⎢⎢⎣⎡ ′′′ φ φ ψ ψ φ φ ψ ψ 0 sin sin 0 cos sin 1 0 0 0 0 1 sin cos 0 cos sin 0 2 2 2 1 256 25 8 5 cos 16 1 5 4 (cos ) n n n n o P P nt P P nt D R + + + ⎟⎟⎠⎞ ⎜⎜⎝⎛ = + + At GEO • Maximum delay length from GEO (x’,D) = 0.310Ro • Minimum semi-minor axis projection (y’,z’) = 0.914Ro Collector s/c trajectory in propagation vector’s (x’) frame: Collector-Combiner s/c distance:

Mission Parameters Components Steered Planar ODL Combiner s/c 1821kg 182.1 Combiner Propellant △WRTm)=056 Collector S/C 871Kg 87.1k Collector Delay Lines 0.34R0 Collector Propellant AV/(n Ro Tfe)=1.55 Spacecraft Mass estimates from For each spacecraft initial Deep Space 3 (DS3)design · Determine△∨ Tfo 5 years Propellant mass from Rocket equation Ro=500 m(DS3-1000 m baseline) Place odl on collector s/c m Ease of operation mo-Propellant Mass(kg) Lower overall dry mass and nd-Spacecraft Dry Mass(kg) therefore, lower system mass sp -Specific impulse( sec) g-Earth's gravity (9.81 m/sec) Space Systems Laboratory Massachusetts Institute of Technology

Space Systems Laboratory Massachusetts Institute of Technology Mission Parameters Mission Parameters Components Steered Planar ODL Combiner S/C 182.1 kg 182.1 kg Combiner Propellant - ∆V/(n2RoTlife) = 0.56 Collector S/C 87.1 kg 87.1 kg Collector Delay Lines - 0.34Ro Collector Propellant ∆V/(n2RoTlife) = 1.55 - Spacecraft Mass estimates from initial Deep Space 3 (DS3) design • Tlife = 5 years • Ro = 500 m (DS3 - 1000 m baseline) For each spacecraft • Determine ∆V • Propellant mass from Rocket equation 1 V exp −⎟⎟⎠⎞ ⎜⎜⎝⎛ ∆ = m I g md sp p mp - Propellant Mass (kg) md - Spacecraft Dry Mass (kg) Isp - 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)

Impact of ODL General observations Relatively insensitive to the number of collector s/c( 4 collector) DLC moul(exp(ar/sp8) Trading between propellant and ODL 0.34R mass 04 R0=500m R=500m 0.35 Break even point in 250 s (DLC=0. 1 kg/m Arch 1: comb=182. 1, macoll =114.19 25 Arch 2: comb=200.4, moll =104.1 50.2 R=50m Break even point Isn 220 s (DLC =0.1 kg/ 0.05 Arch 1: comb=182.1, moll = 89.7 Arch 2: comb =184.1, moll = 88.8 1002003004005006007008009001000 Specific Impulse( secs) Space Systems Laboratory Massachusetts Institute of Technology

Space Systems Laboratory Massachusetts Institute of Technology Impact of ODL Impact of ODL General Observations • Relatively insensitive to the number of collector s/c (> 4 collector) • Trading between propellant and ODL mass Ro = 500 m • Break even point Isp = 250 s (DLC = 0. 1 kg/m) • Arch 1 : mcomb = 182.1, mcoll = 114.1 • Arch 2 : mcomb = 200.4, mcoll = 104.1 Ro = 50 m • Break even point Isp = 220 s (DLC = 0.1 kg/m) • Arch 1 : mcomb = 182.1, mcoll = 89.7 • Arch 2 : mcomb = 184.1, mcoll = 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 Length Convers ion ( kg/ m ) R o = 500 m R o = 50 m (exp( / ) 1) 0.34 coll sp o m V I g DLC R ∆ − ≈