Orbital Mechanics • 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 © 2020 David L. Akin - All rights reserved http://spacecraft.ssl.umd.edu U N I V E R S I T Y O F Orbital Mechanics MARYLAND 1 ENAE 791 - Launch and Entry Vehicle Design Orbital Mechanics: 500 years in 40 min. • Newton’s Law of Universal Gravitation Gm m F = 1 2 r2 • Newton’s First Law meets vector algebra −⇥F = m−⇥a U N I V E R S I T Y O F Orbital Mechanics MARYLAND 2 ENAE 791 - Launch and Entry Vehicle Design Relative Motion Between Two Bodies 2 d r1⃗ m1m2 r21⃗ m2 m = G ⇥r2 1 dt2 2 r21⃗ r21⃗ ⇥ m1m2 m1m2 F12 = G r21⃗ = G (r2⃗ − r1⃗ ) ⇥r1 3 3 F⇥21 r21⃗ r21⃗ 2 d r2⃗ m1m2 m m2 = G (r1⃗ − r2⃗ ) 1 dt2 3 r12⃗ F⇥12 = force due to body 1 on body 2 U N I V E R S I T Y O F Orbital Mechanics MARYLAND 3 ENAE 791 - Launch and Entry Vehicle Design Gravitational Motion Let µ = G(m1 + m2) d2~r ~r + µ = ~0 dt2 r3 “Equation of Orbit” - Orbital motion is simple harmonic motion U N I V E R S I T Y O F Orbital Mechanics MARYLAND 4 ENAE 791 - Launch and Entry Vehicle Design Orbital Angular Momentum ⃗ h is angular momentum vector (constant) =⇒ ⃗r and ⃗v are in a constant plane U N I V E R S I T Y O F Orbital Mechanics MARYLAND 5 ENAE 791 - Launch and Entry Vehicle Design Fun and Games with Algebra 0 U N I V E R S I T Y O F Orbital Mechanics MARYLAND 6 ENAE 791 - Launch and Entry Vehicle Design More Algebra, More Fun U N I V E R S I T Y O F Orbital Mechanics MARYLAND 7 ENAE 791 - Launch and Entry Vehicle Design Orientation of the Orbit ~e eccentricity vector, in orbital plane ⌘ ~e points in the direction of periapsis U N I V E R S I T Y O F Orbital Mechanics MARYLAND 8 ENAE 791 - Launch and Entry Vehicle Design Position in Orbit θ = true anomaly: angular travel from perigee passage U N I V E R S I T Y O F Orbital Mechanics MARYLAND 9 ENAE 791 - Launch and Entry Vehicle Design Relating Velocity and Orbital Elements U N I V E R S I T Y O F Orbital Mechanics MARYLAND 10 ENAE 791 - Launch and Entry Vehicle Design Vis-Viva Equation 1 e2 p a(1 e2)= − ⌘ − 2 v2 r − µ 1 2 v2 − a = r − µ ✓ ◆ 2 1 v2 = µ <--Vis-Viva Equation r − a ✓ ◆ v2 µ µ = 2 − r −2a U N I V E R S I T Y O F Orbital Mechanics MARYLAND 11 ENAE 791 - Launch and Entry Vehicle Design Energy in Orbit • Kinetic Energy 1 K.E. v2 K.E. = mν 2 ⇒ = 2 m 2 • Potential Energy mµ P.E. µ P.E. = − ⇒ = − r m r • Total Energy v2 µ µ Const. = − = − <--Vis-Viva Equation 2 r 2a U N I V E R S I T Y O F Orbital Mechanics MARYLAND 12 ENAE 791 - Launch and Entry Vehicle Design Suborbital Tourism - Spaceship Two U N I V E R S I T Y O F Orbital Mechanics MARYLAND 13 ENAE 791 - Launch and Entry Vehicle Design How Close are we to Space Tourism? • Energy for 100 km vertical climb µ µ km2 MJ + = 0.965 2 = 0.965 −rE + 100 km rE sec kg • Energy for 200 km circular orbit µ µ km2 MJ + = 32.2 2 = 32.2 −2(rE + 200 km) rE sec kg • Energy difference is a factor of 33! U N I V E R S I T Y O F Orbital Mechanics MARYLAND 14 ENAE 791 - Launch and Entry Vehicle Design 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 MARYLAND 15 ENAE 791 - Launch and Entry Vehicle Design 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 MARYLAND 16 ENAE 791 - Launch and Entry Vehicle Design Classical Parameters of Elliptical Orbits U N I V E R S I T Y O F Orbital Mechanics MARYLAND 17 ENAE 791 - Launch and Entry Vehicle Design Basic Orbital Parameters • Semi-latus rectum (or parameter) 2 p = a(1 − e ) • 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 = ⃗r × ⃗v h = √µp U N I V E R S I T Y O F Orbital Mechanics MARYLAND 18 ENAE 791 - Launch and Entry Vehicle Design 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 MARYLAND 19 ENAE 791 - Launch and Entry Vehicle Design The Hohmann Transfer r2 v2 r1 vapogee v1 vperigee U N I V E R S I T Y O F Orbital Mechanics MARYLAND 20 ENAE 791 - Launch and Entry Vehicle Design First Maneuver Velocities • Initial vehicle velocity µ v1 = !r1 • Needed final velocity µ 2r2 vperigee = !r1 !r1 + r2 • Required ΔV µ 2r2 ∆v1 = 1 !r1 "!r1 + r2 − # U N I V E R S I T Y O F Orbital Mechanics MARYLAND 21 ENAE 791 - Launch and Entry Vehicle Design Second Maneuver Velocities • Initial vehicle velocity µ 2r1 vapogee = !r2 !r1 + r2 • Needed final velocity µ v2 = !r2 • Required ΔV µ 2r1 ∆v2 = 1 !r2 " − !r1 + r2 # U N I V E R S I T Y O F Orbital Mechanics MARYLAND 22 ENAE 791 - Launch and Entry Vehicle Design Limitations on Launch Inclinations Equator U N I V E R S I T Y O F Orbital Mechanics MARYLAND 23 ENAE 791 - Launch and Entry Vehicle Design Differences in Inclination Line of Nodes U N I V E R S I T Y O F Orbital Mechanics MARYLAND 24 ENAE 791 - Launch and Entry Vehicle Design Choosing the Wrong Line of Apsides U N I V E R S I T Y O F Orbital Mechanics MARYLAND 25 ENAE 791 - Launch and Entry Vehicle Design Simple Plane Change v1 Δv2 vapogee v vperigee 2 U N I V E R S I T Y O F Orbital Mechanics MARYLAND 26 ENAE 791 - Launch and Entry Vehicle Design Optimal Plane Change v v perigee 1 Δv2 vapogee Δv1 v2 U N I V E R S I T Y O F Orbital Mechanics MARYLAND 27 ENAE 791 - Launch and Entry Vehicle Design First Maneuver with Plane Change Δi1 • Initial vehicle velocity µ v = 1 r 1 • Needed final velocity µ 2r v = 2 p r r + r 1 1 2 • Required ΔV ∆v = v2 + v2 2v v cos ∆i 1 1 p − 1 p 1 U N I V E R S I T Y O F Orbital Mechanics MARYLAND 28 ENAE 791 - Launch and Entry Vehicle Design Second Maneuver with Plane Change Δi2 • Initial vehicle velocity µ 2r v = 1 a r r + r 2 1 2 • Needed final velocity µ v = 2 r 2 • Required ΔV ∆v = v2 + v2 2v v cos ∆i 2 2 a − 2 a 2 U N I V E R S I T Y O F Orbital Mechanics MARYLAND 29 ENAE 791 - Launch and Entry Vehicle Design Sample Plane Change Maneuver Optimum initial plane change = 2.20° U N I V E R S I T Y O F Orbital Mechanics MARYLAND 30 ENAE 791 - Launch and Entry Vehicle Design Calculating Time in Orbit U N I V E R S I T Y O F Orbital Mechanics MARYLAND 31 ENAE 791 - Launch and Entry Vehicle Design Time in Orbit • Period of an orbit a3 P = 2⇥ 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 MARYLAND 32 ENAE 791 - Launch and Entry Vehicle Design 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 MARYLAND 33 ENAE 791 - Launch and Entry Vehicle Design 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 ttransit = = ⇥ = 19, 034 sec = 5h17m14s 2 µ U N I V E R S I T Y O F Orbital Mechanics MARYLAND 34 ENAE 791 - Launch and Entry Vehicle Design Example: Time-based Position Find the spacecraf position 3 hours afer perigee µ 4 rad n = = 1.650x10− a3 sec r e = 1 p = 0.7270 − a Ej+1 = nt + e sin Ej =1.783 + 0.7270 sin Ej 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 MARYLAND 35 ENAE 791 - Launch and Entry Vehicle Design Example: Time-based Position (cont.) 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 ⇥ − Have to be sure to get the position in the proper quadrant - since the time is less than 1/2 the period, the spacecraf has yet to reach apogee --> 0°<θ<180° U N I V E R S I T Y O F Orbital Mechanics MARYLAND 36 ENAE 791 - Launch and Entry Vehicle Design Velocity Components in Orbit p r = 1+e cos θ dr d p p( e sin θ dθ ) v = = = − − dt r dt dt 1+e cos θ (1 + e cos θ)2 ⇥ pe sin θ dθ v = r (1 + e cos θ)2 dt p r2 dθ e sin θ 1+e cos θ = v = dt r ⇒ r p ⇤h = r v ⇤ ⇥ ⇤ U N I V E R S I T Y O F Orbital Mechanics MARYLAND 37 ENAE 791 - Launch and Entry Vehicle Design Velocity Components in Orbit (cont.) U N I V E R S I T Y O F Orbital Mechanics MARYLAND 38 ENAE 791 - Launch and Entry Vehicle Design Patched Conics • Simple approximation to multi-body motion (e.g., traveling from Earth orbit through solar orbit into Martian orbit) • Treats multibody problem as “hand-offs” between gravitating bodies --> reduces analysis to sequential two- body problems • Caveat Emptor: Tere are a number of formal methods to perform patched conic analysis.