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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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  • Mercury's Resonant Rotation from Secular Orbital Elements

    Mercury's Resonant Rotation from Secular Orbital Elements

    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Institute of Transport Research:Publications Mercury’s resonant rotation from secular orbital elements Alexander Stark a, Jürgen Oberst a,b, Hauke Hussmann a a German Aerospace Center, Institute of Planetary Research, D-12489 Berlin, Germany b Moscow State University for Geodesy and Cartography, RU-105064 Moscow, Russia The final publication is available at Springer via http://dx.doi.org/10.1007/s10569-015-9633-4. Abstract We used recently produced Solar System ephemerides, which incorpo- rate two years of ranging observations to the MESSENGER spacecraft, to extract the secular orbital elements for Mercury and associated uncer- tainties. As Mercury is in a stable 3:2 spin-orbit resonance these values constitute an important reference for the planet’s measured rotational pa- rameters, which in turn strongly bear on physical interpretation of Mer- cury’s interior structure. In particular, we derive a mean orbital period of (87.96934962 ± 0.00000037) days and (assuming a perfect resonance) a spin rate of (6.138506839 ± 0.000000028) ◦/day. The difference between this ro- tation rate and the currently adopted rotation rate (Archinal et al., 2011) corresponds to a longitudinal displacement of approx. 67 m per year at the equator. Moreover, we present a basic approach for the calculation of the orientation of the instantaneous Laplace and Cassini planes of Mercury. The analysis allows us to assess the uncertainties in physical parameters of the planet, when derived from observations of Mercury’s rotation. 1 1 Introduction Mercury’s orbit is not inertially stable but exposed to various perturbations which over long time scales lead to a chaotic motion (Laskar, 1989).