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Orbital MechanicsENAE 483/788D - Principles of Space Systems Design

U N I V E R S I T Y O FMARYLAND

Orbital Mechanics• Lecture #03 – September 8, 2011• ENAE 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”)

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© 2011 David L. Akin - All rights reservedhttp://spacecraft.ssl.umd.edu



Notes

• There will be no live lecture on Tuesday, 9/13 - a lecture video will be posted

• Looking ahead - under current plans, the following lectures will also be video only– Tuesday, 9/20– Thursday, 9/22– Thursday, 9/29

• As always, future plans are subject to change at any time, so keep checking the syllabus at spacecraft.ssl.umd.edu

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Exo-SPHERES Project Team• Systems Integration

– Gabriel Charalambides– Raymond Russell– Terry Van Wormer

• Mission Planning and Analysis– Justin Brannan– Josh Fogel– Samuel Lewis

• Loads, Structures, and Mechanisms– Andrei Arevalo– Daniel Skeberdis– Walter McGee

• Power, Propulsion, and Thermal– Henry Elder– Ethan Evans– Angela Maki

• Crew Systems– Pratik Saripalli– Sean Li

• Avionics and Software– Reuben Abraham– Anthony Morales– Sanka Perera– Sean Hersey– Michael Tsu– Benjamin Krosner

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Human/Robotic Servicing Project Team• Systems Integration

– Donald Clabaugh– Bradley Hood– Brian Chinn

• Mission Planning and Analysis– Marco Colleluori– Elizabeth Lato– Lucrecio Alberto

• Loads, Structures, and Mechanisms– David Thoerig– Jason Lu– William Gross

• Power, Propulsion, and Thermal– Richard Burcat– Grant Barrett– Jamil Shehadeh

• Crew Systems– Jason Niemeyer– Sahil Ambani

• Avionics and Software– Matthew Westerfield– Joseph Chung– Alexander Nelson

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Space Launch - The Physics

• Minimum orbital altitude is ~200 km

• Circular orbital velocity there is 7784 m/sec

• Total energy per kg in orbit

Potential Energy

kg in orbit= − µ

rorbit+

µ

rE= 1.9× 106 J

kg

Kinetic Energy

kg in orbit=

12

µ

r2orbit

= 30× 106 J

kg

Total Energy

kg in orbit= KE + PE = 32× 106 J

kg

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Theoretical Cost to Orbit

• Convert to usual energy units

• Domestic energy costs are ~$0.05/kWhr

Theoretical cost to orbit $0.44/kg

Total Energy

kg in orbit= 32× 106 J

kg= 8.888

kWhrs

kg

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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!

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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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Total Energykg

=thrust× distance

mass=

Td

m=

gd

L/D

WeightThrust

=LiftDrag

, ormg

T=

L

D



Equivalent Airline Costs?

• Average economy ticket NY-Sydney round-round-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!

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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 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!

• (If you’re interesting in how this works out, you can take ENAE 791 Launch and Entry Vehicle Design next term...)

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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,

F =

GMm

r2

µ = GM

g =

µ

r2

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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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Energy in Orbit

• Kinetic Energy

• Potential Energy

• Total Energy

<--Vis-Viva Equation

K.E. =1

2mv

2=!

K.E.

m=

v2

2

P.E. = !µm

r="

P.E.

m= !

µ

r

Constant =v2

2!

µ

r= !

µ

2a

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v2 = µ

�2r− 1

a

�



Classical Parameters of Elliptical Orbits

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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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Implications of Vis-Viva

• Circular orbit (r=a)

• Parabolic escape orbit (a tends to infinity)

• Relationship between circular and parabolic orbits

vcircular =

!

µ

r

vescape =

!

2µ

r

vescape =!

2vcircular

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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

v1 =

!

µ

r1

vperigee =

!

µ

r1

!

2r2

r1 + r2

!v1 =

!

µ

r1

"!

2r2

r1 + r2

! 1

#

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Second Maneuver Velocities

• Initial vehicle velocity


• Delta-V

!v2 =

!

µ

r2

"

1 !

!

2r1

r1 + r2

#

vapogee =

!

µ

r2

!

2r1

r1 + r2

v2 =

!

µ

r2

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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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∆VLow Thrust = Vc1 − Vc2 =

�µ

r1−�

µ

r2



Limitations on Launch Inclinations

Equator

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Differences in Inclination

Line of Nodes

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Choosing the Wrong Line of Apsides

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Simple Plane Change

vperigee

v1vapogee

v2

Δv2

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Optimal Plane Change

vperigee v1 vapogee

v2

Δv2Δv1

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First Maneuver with Plane Change Δi1



• Delta-V

v1 =�

µ

r1

vp =�

µ

r1

�2r2

r1 + r2

∆v1 =�

v21 + v2

p − 2v1vp cos ∆i1

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Second Maneuver with Plane Change Δi2



• Delta-V

∆v2 =�

v22 + v2

a − 2v2va cos ∆i2

va =�

µ

r2

�2r1

r1 + r2

v2 =�

µ

r2

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Sample Plane Change Maneuver

Optimum initial plane change = 2.20°

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Calculating Time in Orbit

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Time in Orbit

• Period of an orbit

• Mean motion (average angular velocity)

• Time since pericenter passage

➥M=mean anomaly

P = 2π

�a3

µ

n =�

µ

a3

M = nt = E − e sinE

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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

r = a (1− e cos E)

tanθ

2=

�1 + e

1− etan

E

2

Ei+1 = nt + e sinEi

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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)

a =12

(r1 + r2) = 24, 459 km

ttransit =P

2= π

�a3

µ= 19, 034 sec = 5h17m14s

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Example: Time-based Position

Find 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

Ej+1 = nt + e sin Ej = 1.783 + 0.7270 sin Ej

n =�

µ

a3= 1.650x10−4 rad

sec

e = 1− rp

a= 0.7270

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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

tanθ

2=

�1 + e

1− etan

E

2=⇒ θ = 160 deg

r = a(1− e cosE) = 12, 387 km



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 ! e2)

r =p

1 + e cos !

rp = a(1 ! e) ra = a(1 + e)

h =

!

µp

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Velocity Components in Orbit

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r =p

1 + e cos θ

vr =dr

dt=

d

dt

�p

1 + e cos θ

�=−p(−e sin θ dθ

dt )(1 + e cos θ)2

vr =pe sin θ

(1 + e cos θ)2dθ

dt

1 + e cos θ =p

r⇒ vr =

r2 dθdt e sin θ

p−→h = −→r ×−→v



Velocity Components in Orbit (cont.)

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�h = �r × �v h = rv cos γ = r

�rdθ

dt

�= r2

dθ

dt



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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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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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

v =

!

v2

h+

2µ

r

!v =

!

v2

h+

2µ

r!

!

µ

r

v2

2− µ

r= − µ

2a=

v2h

2

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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)

vm =

!

µ

rm

=

!

398, 604

384, 400= 1.018

km

sec

va = vm

!

2r1

r1 + rm

= 1.018

!

6778

6778 + 384, 400= 0.134

km

sec

vh = vm ! va = vm = 1.018 ! 0.134 = 0.884km

sec

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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

vpm =

!

v2

h +2µm

rLLO=

!

1.0182 +2(4667.9)

1878= 2.451

km

sec

!v = vpm ! vcm = 2.451 !

!

4667.9

1878= 0.874

km

sec

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ΔV Requirements for Lunar Missions

To:

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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LOI ΔV Based on Landing Site

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LOI ΔV Including Loiter Effects

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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

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Hill’s Equations (Proximity Operations)

€

˙ ̇ x = 3n2x + 2n˙ y + adx

€

˙ ̇ y = −2n˙ x + ady

€

˙ ̇ z = −n 2z + adz

Ref: J. E. Prussing and B. A. Conway, Orbital Mechanics Oxford University Press, 1993

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Clohessy-Wiltshire (“CW”) Equations

€

x(t) = 4 − 3cos(nt)[ ]xo +sin(nt)

n˙ x o +

2n

1− cos(nt)[ ] ˙ y o

€

y(t) = 6 sin(nt)− nt[ ]xo + yo −2n

1−cos(nt)[ ] ˙ x o +4sin(nt)− 3nt

n˙ y o

€

z( t) = zo cos(nt) +˙ z on

sin(nt)

€

˙ z ( t) = −zonsin(nt) + ˙ z o sin(nt)

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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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“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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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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