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 Lecture #05 September 15, 2015 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 ) 2015 David L. Akin - All rights reserved

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

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

4 Actual Cost to Orbit SpaceX Falcon 9 13,150 kg to LEO $65 M per flight Lowest cost system currently flying $4940/kg of payload Factor of 11,000x higher than theoretical energy costs

5 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)

6 Equivalent Airline Costs? Average economy ticket NY-Sydney round-roundtrip (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 190x less than current launch costs But you get to refuel at each stop

7 Equivalence to Air Transport 81,000 km ~ twice around the world Voyager - one of only two aircraft to ever circle the world non-stop, nonrefueled - once

8 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...)

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

10 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

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

12 Classical Parameters of Elliptical Orbits

13 The Classical Orbital Elements Ref: J. E. Prussing and B. A. Conway, Oxford University Press, 1993

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

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

16 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

17 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 )

18 Implications of Hohmann Transfers Implicit assumption is made that velocity changes instantaneously - impulsive thrust Decent assumption if acceleration ~5 m/sec 2 (0.5 g Earth ) 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 r 1 and r 2 is V Low T hrust = V c1 V c2 = r µ r 1 r µ r 2

19 Limitations on Launch Inclinations Equator

20 Differences in Inclination Line of Nodes

21 Choosing the Wrong Line of Apsides

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

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

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

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

26 Sample Plane Change Maneuver Optimum initial plane change = 2.20

27 Calculating Time in Orbit

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

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

30 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 = a 3 µ = 19, 034 sec = 5h17m14s

31 Example: Time-based Position Find the spacecraft position 3 hours after perigee n = µ 4 rad = 1.650x10 a3 sec r p e = 1 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; 2.317

32 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 tan 2 = 1+e 1 e tan E 2 = = 160 deg

33 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

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

35 Velocity Components in Orbit (cont.) ~ h = ~r ~v h = rv cos = r r d dt = r 2 d dt

36 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 twobody 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.

37 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

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

39 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

40 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 = r vh 2 + 2µ m = r LLO r (4667.9) 1878 The required delta-v to slow down into low lunar orbit is v = v pm v cm =2.398 r Red text is a correction to original notes =2.398 km sec =0.822 km sec

41 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

42 LOI ΔV Based on Landing Site

43 LOI ΔV Including Loiter Effects

44 Interplanetary Trajectory Types

45 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

46 C3 for Earth-Mars Transfer

47 Earth-Mars Transfer 2033

48 Earth-Mars Transfer 2037

49 Interplanetary Delta-V Hyperbolic excess velocity V h C 3 = V 2 h V req = p V 2 esc + C 3 V = p V 2 esc + C 3 V c 2033 Window: V =3.55 km/sec 2037 Window: V =3.859 km/sec V in departure from 300 km LEO

50 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, Orbital Mechanics Oxford University Press, 1993

51 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 o nsin(nt) + z o sin(nt) [ 1 cos(nt) ] y o 4sin(nt) 3nt n y o

52 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

53 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

54 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, 1993