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

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1 Discussion of 483/484 project organization 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 ) David L. Akin - All rights reserved

2 ENAE 483/483 Project Structure Matrix organization everyone has a Project responsibility (NEO mission/exo-spheres) Disciplinary specialty Specialty groups Avionics and Software Crew Systems Loads, Structures, and Mechanisms Power, Propulsion, and Thermal Mission Planning and Analysis Systems Integration 2

3 Avionics and Software Data management (flight control computers) Sensor networks Communications bus architecture Navigation Guidance Control algorithms Attitude determination and control Software coding and verification Communications 3

4 Crew Systems Habitat layout Habitability assessments Life support systems design Crew-computer interfaces Crew-robot interfaces Extravehicular activity Crew safety (e.g., fire, radiation) 4

5 Loads, Structures, and Mechanisms Determination of loads sources Flight Impact (e.g., docking, collision, landing) Ground handling Structural component design Stress analysis Summary loads table (non-negative margins) Design of actuators, latches, moving components Mounting of all systems (e.g., propulsion, avionics) 5

6 Power, Propulsion, and Thermal Design of propulsion systems Primary (orbital maneuvering) Reaction control system (attitude and fine translation) Power generation Energy storage Power budgets with margin Thermal load tracking Thermal equilibrium calculations 6

7 Mission Planning and Analysis* Mission science design Science instrument specification Design reference mission creation Selection of candidate targets Orbital mechanics *Much of this will be the responsibility of ASU 7

8 Systems Integration Team coordination Systems engineering Development and tracking of production schedules Budgeting (e.g., cost, mass) Configuration control Requirements flow-down Work breakdown structure System block diagrams Vehicle configuration Systems integration (overall layout) 8

9 Space Launch - The Physics Minimum orbital altitude is ~200 km P otential Energy kg in orbit Circular orbital velocity there is 7784 m/sec Kinetic Energy kg in orbit Total energy per kg in orbit = µ r orbit + µ r E = = 1 2 µ r 2 orbit = J kg J kg T otal Energy kg in orbit = KE + P E = J kg 9

10 Theoretical Cost to Orbit Convert to usual energy units T otal Energy kg in orbit = J kg = kw hrs kg Domestic energy costs are ~$0.05/kWhr Theoretical cost to orbit $0.44/kg 10

11 Actual Cost to Orbit Delta IV Heavy 23,000 kg to LEO $250 M per flight $10,900/kg of payload Factor of 25,000x higher than theoretical energy costs! 11

12 What About Airplanes? For an aircraft in level flight, Weight Thrust = Lift mg, or Drag T Energy = force x distance, so = L D Total Energy thrust distance = = Td kg mass m = gd L/D For an airliner (L/D=25) to equal orbital energy, d=81,000 km (2 roundtrips NY-Sydney) 12

13 Equivalent Airline Costs? Average economy ticket NY-Sydney roundround-trip (Travelocity 9/3/09) ~$1300 Average passenger (+ luggage) ~100 kg Two round trips = $26/kg Factor of 60x more than electrical energy costs Factor of 420x less than current launch costs But you get to refuel at each stop! 13

14 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! 14

15 Orbital Entry - The Physics 32 MJ/kg dissipated by friction with atmosphere over ~8 min = 66kW/kg Pure graphite (carbon) high-temperature material: c p =709 J/kg K Orbital energy would cause temperature gain of 45,000 K! (If you re interesting in how this works out, you can take ENAE 791 Launch and Entry Vehicle Design next term...) 15

16 Newton s Law of Gravitation Inverse square law F = GMm Since it s easier to remember one number, r 2 µ = GM If you re looking for local gravitational acceleration, g = µ r 2 16

17 Some Useful Constants Gravitation constant µ = GM Earth: 398,604 km 3 /sec 2 Moon: km 3 /sec 2 Mars: 42,970 km 3 /sec 2 Sun: 1.327x10 11 km 3 /sec 2 Planetary radii r Earth = 6378 km r Moon = 1738 km r Mars = 3393 km 17

18 Energy in Orbit Kinetic Energy K.E. = 1 2 mv2 = K.E. m Potential Energy P.E. = µm r Total Energy = P.E. m = v2 2 = µ r Constant = v2 2 µ r = µ 2a 2 v 2 = µ r 1 a 18 <--Vis-Viva Equation

19 Classical Parameters of Elliptical Orbits 19

20 The Classical Orbital Elements Ref: J. E. Prussing and B. A. Conway, Oxford University Press,

21 Implications of Vis-Viva Circular orbit (r=a) v circular = Parabolic escape orbit (a tends to infinity) v escape = µ r 2µ Relationship between circular and parabolic orbits r v escape = 2v circular 21

22 The Hohmann Transfer r 2 r 1 v 2 v apogee v 1 v perigee 22

23 First Maneuver Velocities Initial vehicle velocity Needed final velocity Delta-V v perigee = v 1 = µ v 1 = r 1 µ µ r 1 r 1 2r2 r 1 + r 2 ( ) 2r2 1 r 1 + r 2 23

24 Second Maneuver Velocities Initial vehicle velocity Needed final velocity Delta-V v apogee = v 2 = µ v 2 = r 2 µ r 2 ( 1 µ r 2 2r1 r 1 + r 2 2r1 r 1 + r 2 ) 24

25 Limitations on Launch Inclinations Equator 25

26 Differences in Inclination Line of Nodes 26

27 Choosing the Wrong Line of Apsides 27

28 Simple Plane Change v 1 v perigee v apogee Δv 2 v 2 28

29 Optimal Plane Change v perigee v 1 Δv v apogee 1 Δv 2 v 2 29

30 First Maneuver with Plane Change Δi 1 Initial vehicle velocity Needed final velocity Delta-V v 1 = v p = v 1 = µ r 1 µ r 1 2r2 r 1 + r 2 v v2 p 2v 1 v p cos i 1 30

31 Second Maneuver with Plane Change Δi 2 Initial vehicle velocity Needed final velocity Delta-V v 2 = v a = µ r 2 v 2 = µ 2r1 r 1 + r 2 r 2 v v2 a 2v 2 v a cos i 2 31

32 Sample Plane Change Maneuver Optimum initial plane change =

33 Bielliptic Transfer Δv 3 Δv 1 Δv 1 Δv 2 Δv 2 33

34 Coplanar Transfer Velocity Requirements Ref: J. E. Prussing and B. A. Conway, Oxford University Press,

35 Noncoplanar Bielliptic Transfers Δv 3 Δv 1 Δv 1 Δv 2 Δv 2 35

36 Calculating Time in Orbit 36

37 Time in Orbit Period of an orbit Mean motion (average angular velocity) Time since pericenter passage M=mean anomaly P = 2π n = µ a 3 a 3 µ M = nt = E e sin E 37

38 Dealing with the Eccentric Anomaly Relationship to orbit Relationship to true anomaly Calculating M from time interval: iterate until it converges r = a (1 e cos E) tan θ 1 + e 2 = 1 e tan E 2 E i+1 = nt + e sin E i 38

39 Example: Time in Orbit Hohmann transfer from LEO to GEO h 1 =300 km --> r 1 = =6678 km r 2 =42240 km Time of transit (1/2 orbital period) a = 1 2 (r 1 + r 2 ) = 24, 459 km t transit = P 2 = π a 3 µ = 19, 034 sec = 5h17m14s 39

40 Example: Time-based Position Find the spacecraft position 3 hours after perigee µ rad n = = 1.650x10 4 a3 sec e = 1 r p a = E j+1 = nt + e sin E j = sin E j 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;

41 Example: Time-based Position (cont.) E =2.317 r = a(1 e cos E) = 12, 387 km tan θ 1+e 2 = 1 e tan E = θ = 160 deg 2 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 41

42 Basic Orbital Parameters Semi-latus rectum (or parameter) Radial distance as function of orbital position Periapse and apoapse distances Angular momentum h = r v p = a(1 e 2 ) r = p 1+e cos θ r p = a(1 e) r a = a(1 + e) h = µp 42

43 Velocity Components in Orbit v r = dr dt = d dt p r = 1+ecos θ p = 1+ecos θ v r = pe sin θ dθ (1 + e cos θ) 2 dt 1+e cos θ = p r v r = h = r v p( e sin θ dθ dt ) (1 + e cos θ) 2 r2 dθ dt e sin θ p 43

44 Velocity Components in Orbit (cont.) h = r v h = rv cos γ = r r dθ dt = r 2 dθ dt 44

45 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. 45

46 Example: Lunar Orbit Insertion v 2 is velocity of moon around Earth Moon overtakes spacecraft with velocity of (v 2 -v apogee ) 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 46

47 Planetary Approach Analysis Spacecraft has v h hyperbolic excess velocity, which fixes total energy of approach orbit v 2 2 µ r = µ 2a = v2 h 2 Vis-viva provides velocity of approach v = v 2 h + 2µ r Choose transfer orbit such that approach is tangent to desired final orbit at periapse v = vh 2 + 2µ µ r r 47

48 Patched Conic - Lunar Approach Lunar orbital velocity around the Earth Apogee velocity of Earth transfer orbit from initial 400 km low Earth orbit v a = v m v m = Velocity difference between spacecraft infinitely far away and moon (hyperbolic excess velocity) µ r m = 2r1 r 1 + r m = , , 400 =1.018 km sec , 400 =0.134 km sec v h = v m v a = v m = = km sec 48

49 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 v pm = v 2 h + 2µ m r LLO = The required delta-v to slow down into low lunar orbit is v = v pm v cm = (4667.9) =2.451 km sec =0.874 km sec 49

50 ΔV Requirements for Lunar Missions From: To: Low Earth Orbit Lunar Transfer Orbit Low Lunar Orbit Lunar Descent Orbit Lunar Landing Low Earth Orbit km/sec Lunar Transfer Orbit km/sec km/sec km/sec Low Lunar Orbit km/sec km/sec Lunar Descent Orbit km/sec km/sec Lunar Landing km/sec km/sec 50

51 LOI ΔV Based on Landing Site 51

52 LOI ΔV Including Loiter Effects 52

53 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 53

54 Hill s Equations (Proximity Operations) x = 3n 2 x + 2n y + a dx y = 2n x + a dy z = n 2 z + a dz Ref: J. E. Prussing and B. A. Conway, Oxford University Press,

55 Clohessy-Wiltshire ( CW ) Equations x(t) = [ 4 3cos(nt) ]x o + sin(nt) n x o + 2 n y(t) = 6[ sin(nt) nt]x o + y o 2 n [ 1 cos(nt) ] x o + z(t) = z o cos(nt) + z o n sin(nt) z (t) = z nsin(nt) + z sin(nt) o o [ 1 cos(nt) ] y o 4sin(nt) 3nt n y o 55

56 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,

57 R-Bar Approach Approach from along the radius vector ( Rbar ) Gravity gradients decelerate spacecraft approach velocity - low contamination approach Used for Mir, ISS docking approaches 57 Ref: Collins, Meissinger, and Bell, Small Orbit Transfer Vehicle (OTV) for On-Orbit Satellite Servicing and Resupply, 15th USU Small Satellite Conference, 2001

58 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, Oxford University Press,