Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design
U N I V E R S I T Y O FMARYLAND
Orbital Mechanics• Lecture #04 – September 10, 2020 • 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”) • Low-thrust trajectories
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© 2020 David L. Akin - All rights reserved http://spacecraft.ssl.umd.edu
Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design
U N I V E R S I T Y O FMARYLAND
Space Launch - The Physics• Minimum orbital altitude is ~200 km
• Circular orbital velocity there is 7784 m/sec
• Total energy per kg in orbit
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Potential Energykg in orbit
= −μ
rorbit+
μrE
= 1.9 × 106 Jkg
Kinetic Energykg in orbit
=12
μr2orbit
= 30 × 106 Jkg
Total Energykg in orbit
= KE + PE = 32 × 106 Jkg
Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design
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Theoretical Cost to Orbit• Convert to usual energy units
• Domestic energy costs are ~$0.11/kWhr
!Theoretical cost to orbit $1/kg
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Total Energykg in orbit
= 32 × 106 Jkg
= 8.9kWhrs
kg
Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design
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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!
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Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design
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What About Airplanes?• For an aircraft in level flight,
• Energy = force x distance, so
• For an airliner (L/D=25) to equal orbital energy, d=81,000 km (2 roundtrips NY-Sydney)
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WeightThrust
=Lift
Drag, or
mgT
=LD
Total Energykg
=thrust × distance
mass=
Tdm
=gd
L/D
Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design
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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!
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Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design
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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!
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Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design
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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)
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Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design
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Newton’s Law of Gravitation• Inverse square law
• Since it’s easier to remember one number,
• If you’re looking for local gravitational acceleration,
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F =GMm
r2
μ ≡ GM
g =μr2
Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design
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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
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Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design
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Energy in Orbit
<--Vis-Viva Equation
• Kinetic Energy
• Potential Energy
• Total Energy
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Ekinetic ≡12
mv2 ⟹Ekinetic
m=
12
v2
Epotential ≡ −μmr
⟹Epotential
m= −
μr
Etotal = constant =v2
2−
μr
= −μ2a
v2 = μ ( 2r
−1a )
Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design
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Classical Parameters of Elliptical Orbits
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θ
Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design
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The Classical Orbital Elements
Ref: J. E. Prussing and B. A. Conway, Orbital Mechanics Oxford University Press, 1993
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Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design
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Implications of Vis-Viva• Circular orbit (r=a)
• Parabolic escape orbit (a tends to infinity)
• Relationship between circular and parabolic orbits
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vcircular =μr
vescape =2μr
vescape = 2vcircular
Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design
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The Hohmann Transfer
vperigee
v1
vapogee
v2r1
r2
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First Maneuver Velocities• Initial vehicle velocity
• Needed final velocity
• Delta-V
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v1 =μr1
vperigee =μr1
2r2
r1 + r2
Δv1 = vperigee − v1 =μr1
2r2
r1 + r2− 1
Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design
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Second Maneuver Velocities• Initial vehicle velocity
• Needed final velocity
• Delta-V
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v2 =μr2
vapogee =μr2
2r1
r1 + r2
Δv2 = v2 − vapogee =μr2
1 −2r1
r1 + r2
Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design
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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
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Δvlow thrust = vc1 − vc2 =μr1
−μr2
Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design
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Limitations on Launch Inclinations
Equator
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Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design
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Differences in Inclination
Line of Nodes
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Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design
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Choosing the Wrong Line of Apsides
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Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design
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Simple Plane Change
vperigee
v1vapogee
v2
Δv2
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Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design
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Optimal Plane Change
vperigee v1 vapogeev2
Δv2Δv1
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Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design
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First Maneuver with Plane Change Δi1
• Initial vehicle velocity
• Needed final velocity
• Delta-V
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v1 =μr1
vp =μr1
2r2
r1 + r2
Δv1 = v21 + v2
p − 2v1vp cos Δi1
Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design
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Second Maneuver with Plane Change • Initial vehicle velocity
• Needed final velocity
• Delta-V
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v2 =μr2
va =μr2
2r1
r1 + r2
Δv2 = v22 + v2
a − 2v2va cos Δi2
Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design
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Sample Plane Change Maneuver
Optimum initial plane change = 2.20°
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Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design
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Calculating Time in Orbit
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Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design
U N I V E R S I T Y O FMARYLAND
Time in Orbit• Period of an orbit
• Mean motion (average angular velocity)
• Time since pericenter passage
➥M=mean anomaly
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M ≡ nt = E − e sin E
n =μa3
P = 2πa3
μ
Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design
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Dealing with the Eccentric Anomaly• Relationship to orbit
• Relationship to true anomaly
• Calculating M from time interval: iterate
until it converges
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Ei+1 = nt + e sin Ei
r = a(1 − e cos E)
tanθ2
=1 + e1 − e
tanE2
Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design
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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)
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a =12
(r1 + r2) = 24,459 km
ttransit =P2
= πa3
μ− 19,034 sec = 5h17m14s
Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design
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Example: Time-based PositionFind the spacecraft position 3 hours after perigee
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
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Ei+1 = nt + e sin Ei = 1.783 + 0.7270 sin Ei
n =μa3
= 1.650 × 10−4 radsec
e = 1 −rp
a= 0.7270
Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design
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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°
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E = 2.317
r = a(1 − e cos E) = 12,387 km
tanθ2
=1 + e1 − e
tanE2
⟹ θ = 160 deg
Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design
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Basic Orbital Parameters• Semi-latus rectum (or parameter)
• Radial distance as function of orbital position
• Periapse and apoapse distances
• Angular momentum
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p = a (1 − e2)
r =p
1 + e cos θ
rp = a(1 − e)
h = r × v
ra = a(1 + e)
h = μp
Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design
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Velocity Components in Orbit
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h = r × v
r =p
1 + e cos θ
vr =drdt
=ddt ( p
1 + e cos θ ) =−p (−e sin θ dθ
dt )(1 + e cos θ)2
vr =pe sin θ
(1 + e cos θ)2
dθdt
1 + e cos θ =pr
⇒ vr =r2 dθ
dte sin θ
p
Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design
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Velocity Components in Orbit (cont.)
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vr =r2 dθ
dte sin θ
p=
he sin θp
=pμ
pe sin θ
vr =μp
e sin θ
vθ = rdθdt
= rhr2
=hr
=pμ
rvθ =
μp
(1 + e cos θ)
tan γ =vr
vθ=
e sin θ1 + e cos θ
h = r × v h = rv cos γ = r (rdθdt ) = r2 dθ
dt
Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design
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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.
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Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design
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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”
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Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design
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Planetary Approach Analysis• Spacecraft has vh hyperbolic excess velocity,
which fixes total energy of approach orbit
• Vis-viva provides velocity of approach
• Choose transfer orbit such that approach is tangent to desired final orbit at periapse
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v2
2−
μr
= −μ2a
=v2
h
2
v = v2h +
2μr
Δv = v2h +
2μr
−μr
Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design
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Patched Conic - Lunar Approach• Lunar orbital velocity around the Earth
• Apogee velocity of Earth transfer orbit from initial 400 km low Earth orbit
• Velocity difference between spacecraft “infinitely” far away and moon (hyperbolic excess velocity)
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vm =μrm
=398,604384,400
= 1.018kmsec
va = vm2r1
r1 + rm= 1.018
67786778 + 384,400
= 0.134kmsec
vh = vm − va = vm = 1.018 − 0.134 = 0.884kmsec
Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design
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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
• The required delta-V to slow down into low lunar orbit is
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vpm = v2h +
2μm
rLLO= 0.8842 +
2(4667.9)1878
= 2.398kmsec
Δv = vpm − vcm = 2.398 −4667.91878
= 0.822kmsec
Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design
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ΔV Requirements for Lunar MissionsTo:
From:
Low EarthOrbit
LunarTransferOrbit
Low LunarOrbit
LunarDescentOrbit
LunarLanding
Low EarthOrbit
3.107km/sec
LunarTransferOrbit
3.107km/sec
0.837km/sec
3.140km/sec
Low LunarOrbit
0.837km/sec
0.022km/sec
LunarDescentOrbit
0.022km/sec
2.684km/sec
LunarLanding
2.890km/sec
2.312km/sec
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Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design
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LOI ΔV Based on Landing Site
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Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design
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LOI ΔV Including Loiter Effects
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Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design
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Lagrange Points
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Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design
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Non-Keplerian Orbits
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Whitley and Martinez, “Options for Staging Orbits in Cis-lunar Space” IEEE Aerospace Conference, 2016
Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design
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Transfer to/from NRHO
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Whitley and Martinez, “Options for Staging Orbits in Cis-lunar Space” IEEE Aerospace Conference, 2016
Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design
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Delta-V Costs for Lunar Orbits
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Whitley and Martinez, “Options for Staging Orbits in Cis-lunar Space” IEEE Aerospace Conference, 2016
Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design
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Interplanetary Trajectory Types
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Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design
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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
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Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design
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C3 for Earth-Mars Transfer 1990-2045
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Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design
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Earth-Mars Transfer 2033
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Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design
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Earth-Mars Transfer 2037
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Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design
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Interplanetary Delta-V
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Δv for departure from 300 km orbit
Hyperbolic excess velocity ≡ Vh
C3 = V2h
Vreq = V2esc + C3
ΔV = V2esc + C3 − Vc
2033 window : ΔV = 3.55 km /sec
2037 window : ΔV = 3.859 km /sec
Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design
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Low-Thrust Trajectories• Acceleration measured in milligees • Circular orbits remain circular • Vehicle spirals outbound over many orbits
• From any circular orbit, to escape is
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�v
Δv = vc1 − vc2
Δv =μr
Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design
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Low-Thrust ∆V to Moon• From LEO to Moon’s orbit around Earth
• From Moon’s orbit to low lunar orbit (LLO)
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ΔvE =μE
rLEO−
μE
rMoon
ΔvM =μM
rLLO
Δvtotal = ΔvE + ΔvM
Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design
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Low-Thrust ∆V to Mars• From LEO to Moon’s orbit around Earth
• From Moon’s orbit to low lunar orbit (LLO)
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Δvtotal = ΔvEarth escape + ΔvEarth−Mars + ΔvMars capture
ΔvMars capture =μMars
rLMO
ΔvEarth escape =μEarth
rLEO
ΔvEarth−Mars =μSun
rEarth orbit−
μSun
rMars orbit
Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design
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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
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Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design
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Hill’s Equations (Proximity Operations)
Ref: J. E. Prussing and B. A. Conway, Orbital Mechanics Oxford University Press, 1993
··z = − n2z + adz
··x = 3n2x + 2n ·y + adx
··y = − 2n ·x + ady
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Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design
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Clohessy-Wiltshire (“CW”) Equations 1
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x(t) = [4 − 3 cos(nt)] x0 +sin(nt)
n·x0 +
2n [1 − cos(nt)] ·y0
y(t) = 6 [sin(nt) − nt] x0 −2n [1 − cos(nt)] ·x0 +
4 sin(nt) − 3ntn
·y0
z(t) = cos(nt)z0 +sin(nt)
n·z0
Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design
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Clohessy-Wiltshire (“CW”) Equations 2
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·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
Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design
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“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
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Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design
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“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
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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
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Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design
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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
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