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 Velocity Components in Orbit p r = 1 + e cos θ −p −e sin θ dθ dr d p ( dt ) v = = = r dt dt ( 1 + e cos θ ) (1 + e cos θ)2

pe sin θ dθ v = r (1 + e cos θ)2 dt 2 dθ p r e sin θ 1 + e cos θ = ⇒ v = dt r r p h ⃗ = r⃗ × v ⃗ U N I V E R S I T Y O F Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design MARYLAND 34 Velocity Components in Orbit (cont.) dθ dθ h ⃗ = r⃗ × v ⃗ h = rv cos γ = r r = r2 ( dt ) dt 2 dθ r e sin θ he sin θ pμ v = dt = = e sin θ r p p p μ v = e sin θ r p dθ h h pμ μ v = r = r = = vθ = (1 + e cos θ) θ dt r2 r r p v e sin θ tan γ = r = vθ 1 + e cos θ U N I V E R S I T Y O F Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design MARYLAND 35 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: There are a number of formal methods to perform patched conic analysis. The approach presented here is a very simple, convenient, and not altogether accurate method for performing this calculation. Results will be accurate to a few percent, which is adequate at this level of design analysis. U N I V E R S I T Y O F Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design MARYLAND 36 Example: Lunar Orbit Insertion

• v2 is velocity of moon around Earth • Moon overtakes spacecraft with velocity of (v2-vapogee) • This is the velocity of the spacecraft relative to the moon while it is effectively “infinitely” far away (before lunar gravity accelerates it) = “hyperbolic excess velocity” U N I V E R S I T Y O F Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design MARYLAND 37 Planetary Approach Analysis

• Spacecraft has vh hyperbolic excess velocity, which fixes total energy of approach orbit v2 μ μ v2 − = − = h 2 r 2a 2 • Vis-viva provides velocity of approach 2μ v = v2 + h r • Choose transfer orbit such that approach is tangent to desired final orbit at periapse 2μ μ Δv = v2 + − h r r U N I V E R S I T Y O F Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design MARYLAND 38 Patched Conic - Lunar Approach • Lunar orbital velocity around the Earth μ 398,604 km vm = = = 1.018 rm 384,400 sec • Apogee velocity of Earth transfer orbit from initial 400 km low Earth orbit

2r1 6778 km va = vm = 1.018 = 0.134 r1 + rm 6778 + 384,400 sec • Velocity difference between spacecraft “infinitely” far away and moon (hyperbolic excess velocity) km vh = vm − va = vm = 1.018 − 0.134 = 0.884 U N I V E R S I T Y O F Orbitalse Mechanicsc ENAE 483/788D - Principles of Space Systems Design MARYLAND 39 Patched Conic - Lunar Orbit Insertion • The spacecraft is now in a hyperbolic orbit of the moon. The velocity it will have at the perilune point tangent to the desired 100 km low lunar orbit is 2 2μm 2 2(4667.9) km vpm = vh + = 0.884 + = 2.398 rLLO 1878 sec • The required delta-V to slow down into low lunar orbit is 4667.9 km Δv = v − v = 2.398 − = 0.822 pm cm 1878 sec

U N I V E R S I T Y O F Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design MARYLAND 40 ΔV Requirements for Lunar Missions

To: Low Earth Lunar Low Lunar Lunar Lunar Orbit Transfer Orbit Descent Landing From: Orbit Orbit Low Earth 3.107 Orbit km/sec

Lunar 3.107 0.837 3.140 Transfer km/sec km/sec km/sec Orbit Low Lunar 0.837 0.022 Orbit km/sec km/sec

Lunar 0.022 2.684 Descent km/sec km/sec Orbit Lunar 2.890 2.312 Landing km/sec km/sec

U N I V E R S I T Y O F Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design MARYLAND 41 LOI ΔV Based on Landing Site

U N I V E R S I T Y O F Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design MARYLAND 42 LOI ΔV Including Loiter Effects

U N I V E R S I T Y O F Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design MARYLAND 43 Lagrange Points

U N I V E R S I T Y O F Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design MARYLAND 44 Non-Keplerian Orbits

U N I V E R S I T Y O F Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design MARYLAND 45 Transfer to/from NRHO

U N I V E R S I T Y O F Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design MARYLAND 46 Delta-V Costs for Lunar Orbits

U N I V E R S I T Y O F Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design MARYLAND 47 Interplanetary Trajectory Types

U N I V E R S I T Y O F Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design MARYLAND 48 Interplanetary “Pork Chop” Plots • Summarize a number of critical parameters – Date of departure – Date of arrival – Hyperbolic energy (“C3”) – Transfer geometry • Launch vehicle determines available C3 based on window, payload mass • Calculated using Lambert’s Theorem U N I V E R S I T Y O F Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design MARYLAND 49 C3 for Earth-Mars Transfer 1990-2045

U N I V E R S I T Y O F Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design MARYLAND 50 Earth-Mars Transfer 2033

U N I V E R S I T Y O F Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design MARYLAND 51 Earth-Mars Transfer 2037

U N I V E R S I T Y O F Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design MARYLAND 52 Interplanetary Delta-V

Hyperbolic excess velocity ≡ Vh 2 C3 = Vh

2 Vreq = Vesc + C3

2 ΔV = Vesc + C3 − Vc Δv for departure from 300 km orbit 2033 window : ΔV = 3.55 km/sec 2037 window : ΔV = 3.859 km/sec U N I V E R S I T Y O F Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design MARYLAND 53 Low-Thrust Trajectories • Acceleration measured in milligees • Circular orbits remain circular • Vehicle spirals outbound over many orbits

Δv = vc1 − vc2

• From any circular orbit, v to escape is

μ Δv = r

U N I V E R S I T Y O F Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design MARYLAND 54 Low-Thrust ∆V to Moon • From LEO to Moon’s orbit around Earth

μE μE ΔvE = − rLEO rMoon

• From Moon’s orbit to low lunar orbit (LLO)

μM ΔvM = rLLO

Δvtotal = ΔvE + ΔvM

U N I V E R S I T Y O F Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design MARYLAND 55 Low-Thrust ∆V to Mars • From LEO to Moon’s orbit around Earth μEarth ΔvEarth escape = rLEO μSun μSun ΔvEarth−Mars = − rEarth orbit rMars orbit • From Moon’s orbit to low lunar orbit (LLO)

μMars ΔvMars capture = rLMO

Δvtotal = ΔvEarth escape + ΔvEarth−Mars + ΔvMars capture

U N I V E R S I T Y O F Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design MARYLAND 56 Low-Thrust Caveats • This analysis assumes infinitesimal thrust • ∆V calculation should be conservative for any higher thrust level • This will serve for preliminary analysis of low- thrust mission requirements • More accurate ∆V (and time of flight) calculations will depend on integrating the equations of motion and iterating to match boundary conditions

U N I V E R S I T Y O F Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design MARYLAND 57 Hill’s Equations (Proximity Operations)

·· 2 · x = 3n x + 2ny + adx

·· · y = − 2nx + ady

·· 2 z = − n z + adz 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 58 Clohessy-Wiltshire (“CW”) Equations 1

sin(nt) 2 x(t) = [4 − 3 cos(nt)] x + x· + [1 − cos(nt)] y· 0 n 0 n 0

2 4 sin(nt) − 3nt y(t) = 6 [sin(nt) − nt] x − [1 − cos(nt)] x· + y· 0 n 0 n 0

sin(nt) z(t) = cos(nt)z + z· 0 n 0

U N I V E R S I T Y O F Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design MARYLAND 59 Clohessy-Wiltshire (“CW”) Equations 2

· · · x(t) = 3n sin(nt)x0 + cos(nt)x0 + 2 sin(nt)y0

· · · y(t) = − 6n [1 − cos(nt)] x0 − 2 sin(nt)x0 + [4 cos(nt) − 3] y0

· · z(t) = − n sin(nt)z0 + cos(nt)z0

U N I V E R S I T Y O F Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design MARYLAND 60 “V-Bar” Approach

Ref: Collins, Meissinger, and Bell, Small Orbit Transfer Vehicle (OTV) for On- Orbit Satellite Servicing and Resupply, 15th USU Small Satellite Conference, 2001

U N I V E R S I T Y O F Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design MARYLAND 61 “R-Bar” Approach • Approach from along the radius vector (“R-bar”) • Gravity gradients decelerate spacecraft approach velocity - low contamination approach • Used for Mir, ISS docking approaches

Ref: Collins, Meissinger, and Bell, Small Orbit Transfer Vehicle (OTV) for On-Orbit Satellite Servicing and Resupply, 15th USU Small Satellite Conference, 2001 U N I V E R S I T Y O F Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design MARYLAND 62 Today’s Tools • Basic physics of orbital mechanics • DV requirements for simple orbit changes • Orbital maneuvering with plane changes • Time in orbits • Low-thrust trajectories • Interplanetary trajectories • Linearized relative orbital motion

U N I V E R S I T Y O F Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design MARYLAND 63 References for This Lecture • Wernher von Braun, The Mars Project University of Illinois Press, 1962 • William Tyrrell Thomson, Introduction to Space Dynamics Dover Publications, 1986 • Francis J. Hale, Introduction to Space Flight Prentice-Hall, 1994 • William E. Wiesel, Spaceflight Dynamics MacGraw-Hill, 1997 • 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 64