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Orbital Mechanics Orbital Mechanics • Lecture #04 – September 5, 2019 • Planetary launch and entry overview • Energy and velocity in orbit • Elliptical orbit parameters • Orbital elements • Coplanar orbital transfers • Noncoplanar transfers • Time in orbit • Interplanetary trajectories • Relative orbital motion (“proximity operations”) © 2019 David L. Akin - All rights reserved • Low-thrust trajectories http://spacecraft.ssl.umd.edu U N I V E R S I T Y O F Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design MARYLAND 1 Space Launch - The Physics • Minimum orbital altitude is ~200 km Potential Energy μ μ J = − + = 1.9 × 106 kg in orbit rorbit rE kg • Circular orbital velocity there is 7784 m/sec Kinetic Energy 1 μ J = = 30 × 106 2 kg in orbit 2 rorbit kg • Total energy per kg in orbit Total Energy J = KE + PE = 32 × 106 kg in orbit kg U N I V E R S I T Y O F Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design MARYLAND 2 Theoretical Cost to Orbit • Convert to usual energy units Total Energy J kWhrs = 32 × 106 = 8.9 kg in orbit kg kg • Domestic energy costs are ~$0.11/kWhr !Theoretical cost to orbit $1/kg U N I V E R S I T Y O F Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design MARYLAND 3 Actual Cost to Orbit • SpaceX Falcon 9 – 22,800 kg to LEO – $62 M per flight – Lowest cost system currently flying • $2720/kg of payload • Factor of 2700x higher than theoretical energy costs! U N I V E R S I T Y O F Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design MARYLAND 4 What About Airplanes? • For an aircraft in level flight, Weight Lift mg L = , or = Thrust Drag T D • Energy = force x distance, so Total Energy thrust × distance Td gd = = = kg mass m L/D • For an airliner (L/D=25) to equal orbital energy, d=81,000 km (2 roundtrips NY-Sydney) U N I V E R S I T Y O F Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design MARYLAND 5 Equivalent Airline Costs? • Average economy ticket NY-Sydney round-trip (Travelocity 5/28/19) ~$1000 • Average passenger (+ luggage) ~100 kg • Two round trips = $20/kg – Factor of 20x more than electrical energy costs – Factor of 136x less than current launch costs • But… you get to refuel at each stop! U N I V E R S I T Y O F Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design MARYLAND 6 Equivalence to Air Transport • 81,000 km ~ twice around the world • Voyager - one of only two aircraft to ever circle the world non-stop, non- refueled - once! U N I V E R S I T Y O F Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design MARYLAND 7 Orbital Entry - The Physics • 32 MJ/kg dissipated by friction with atmosphere over ~8 min = 66kW/kg • Pure graphite (carbon) high-temperature material: cp=709 J/kg°K • Orbital energy would cause temperature gain of 45,000°K! • Thus proving the comment about space travel, “It’s utter bilge!” (Sir Richard Wooley, Astronomer Royal of Great Britain, 1956) U N I V E R S I T Y O F Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design MARYLAND 8 Newton’s Law of Gravitation • Inverse square law GMm F = r2 • Since it’s easier to remember one number, μ ≡ GM • If you’re looking for local gravitational acceleration, μ g = r2 U N I V E R S I T Y O F Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design MARYLAND 9 Some Useful Constants • Gravitation constant µ = GM – Earth: 398,604 km3/sec2 – Moon: 4667.9 km3/sec2 – Mars: 42,970 km3/sec2 – Sun: 1.327x1011 km3/sec2 • Planetary radii – rEarth = 6378 km – rMoon = 1738 km – rMars = 3393 km U N I V E R S I T Y O F Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design MARYLAND 10 Energy in Orbit • Kinetic Energy 1 E 1 E ≡ mv2 ⟹ kinetic = v2 kinetic 2 m 2 • Potential Energy μm Epotential μ E ≡ − ⟹ = − potential r m r • Total Energy v2 μ μ E = constant = − = − total 2 r 2a 2 1 v2 = μ − <--Vis-Viva Equation ( r a ) U N I V E R S I T Y O F Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design MARYLAND 11 Classical Parameters of Elliptical Orbits θ U N I V E R S I T Y O F Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design MARYLAND 12 The Classical Orbital Elements Ref: J. E. Prussing and B. A. Conway, Orbital Mechanics Oxford University Press, 1993 U N I V E R S I T Y O F Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design MARYLAND 13 Implications of Vis-Viva • Circular orbit (r=a) μ v = circular r • Parabolic escape orbit (a tends to infinity) 2μ v = escape r • Relationship between circular and parabolic orbits vescape = 2vcircular U N I V E R S I T Y O F Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design MARYLAND 14 The Hohmann Transfer r2 v2 r1 vapogee v1 vperigee U N I V E R S I T Y O F Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design MARYLAND 15 First Maneuver Velocities • Initial vehicle velocity μ v1 = r1 • Needed final velocity μ 2r2 vperigee = r1 r1 + r2 • Delta-V μ 2r2 Δv1 = vperigee − v1 = − 1 r1 r1 + r2 U N I V E R S I T Y O F Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design MARYLAND 16 Second Maneuver Velocities • Initial vehicle velocity μ 2r1 vapogee = r2 r1 + r2 • Needed final velocity μ v2 = r2 (blue color means corrected since lecture) • Delta-V μ 2r1 Δv2 = v2 − vapogee = 1 − r2 r1 + r2 U N I V E R S I T Y O F Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design MARYLAND 17 Implications of Hohmann Transfers • Implicit assumption is made that velocity changes instantaneously - “impulsive thrust” • Decent assumption if acceleration ≥ ~5 m/sec2 (0.5 gEarth) • Lower accelerations result in altitude change during burn ⇒ lower efficiencies and higher ΔVs • Worst case is continuous “infinitesimal” thrusting (e.g., ion engines) ⇒ ΔV between circular coplanar orbits r1 and r2 is μ μ Δvlow thrust = vc1 − vc2 = − r1 r2 U N I V E R S I T Y O F Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design MARYLAND 18 Limitations on Launch Inclinations Equator U N I V E R S I T Y O F Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design MARYLAND 19 Differences in Inclination Line of Nodes U N I V E R S I T Y O F Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design MARYLAND 20 Choosing the Wrong Line of Apsides U N I V E R S I T Y O F Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design MARYLAND 21 Simple Plane Change v1 Δv2 vapogee v vperigee 2 U N I V E R S I T Y O F Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design MARYLAND 22 Optimal Plane Change vperigee v1 Δv2 vapogee Δv1 v2 U N I V E R S I T Y O F Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design MARYLAND 23 First Maneuver with Plane Change Δi1 • Initial vehicle velocity μ v1 = r1 • Needed final velocity μ 2r2 vp = r1 r1 + r2 • Delta-V 2 2 Δv1 = v1 + vp − 2v1vp cos Δi1 U N I V E R S I T Y O F Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design MARYLAND 24 Second Maneuver with Plane Change • Initial vehicle velocity μ 2r1 va = r2 r1 + r2 • Needed final velocity μ v2 = r2 • Delta-V 2 2 Δv2 = v2 + va − 2v2va cos Δi2 U N I V E R S I T Y O F Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design MARYLAND 25 Sample Plane Change Maneuver Optimum initial plane change = 2.20° U N I V E R S I T Y O F Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design MARYLAND 26 Calculating Time in Orbit U N I V E R S I T Y O F Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design MARYLAND 27 Time in Orbit • Period of an orbit a3 P = 2π μ • Mean motion (average angular velocity) μ n = a3 • Time since pericenter passage M ≡ nt = E − e sin E ➥M=mean anomaly U N I V E R S I T Y O F Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design MARYLAND 28 Dealing with the Eccentric Anomaly • Relationship to orbit r = a(1 − e cos E) • Relationship to true anomaly θ 1 + e E tan = tan 2 1 − e 2 • Calculating M from time interval: iterate Ei+1 = nt + e sin Ei until it converges U N I V E R S I T Y O F Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design MARYLAND 29 Example: Time in Orbit • Hohmann transfer from LEO to GEO – h1=300 km --> r1=6378+300=6678 km – r2=42240 km • Time of transit (1/2 orbital period) 1 a = (r + r ) = 24,459 km 2 1 2 P a3 t = = π − 19,034 sec = 5h17m14s transit 2 μ U N I V E R S I T Y O F Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design MARYLAND 30 Example: Time-based Position Find the spacecraft position 3 hours after perigee μ rad n = = 1.650 × 10−4 a3 sec rp e = 1 − = 0.7270 a Ei+1 = nt + e sin Ei = 1.783 + 0.7270 sin Ei E=0; 1.783; 2.494; 2.222; 2.361; 2.294; 2.328; 2.311; 2.320; 2.316; 2.318; 2.317; 2.317; 2.317 U N I V E R S I T Y O F Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design MARYLAND 31 Example: Time-based Position (cont.) Have to be sure to get the position in the proper quadrant - since the time is less than 1/2 the period, the spacecraft has yet to reach apogee -- > 0°<θ<180° E = 2.317 r = a(1 − e cos E) = 12,387 km θ 1 + e E tan = tan ⟹ θ = 160 deg 2 1 − e 2 U N I V E R S I T Y O F Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design MARYLAND 32 Basic Orbital Parameters • Semi-latus rectum (or parameter) p = a (1 − e2) • Radial distance as function of orbital position p r = 1 + e cos θ • Periapse and apoapse distances rp = a(1 − e) ra = a(1 + e) • Angular momentum h ⃗ = r⃗ × v ⃗ h = μp U N I V E R S I T Y O F Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design MARYLAND 33
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