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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
Orbital Mechanics
• 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”)
Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design
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• Kinetic Energy
!• Potential Energy!
• Total Energy
Energy in Orbit
<--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
⇥
Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design
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Classical Parameters of Elliptical Orbits
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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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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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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
∆v2 =
!
µ
r2
"
1 −
!
2r1
r1 + r2
#
vapogee =
!
µ
r2
!
2r1
r1 + r2
v2 =
!
µ
r2
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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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�VLow Thrust
= Vc1 � V
c2 =
rµ
r1�r
µ
r2
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 vapogee
v2
Δv2Δv1
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First Maneuver with Plane Change Δi1
• Initial vehicle velocity !!
• Needed final velocity !!
• 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
• Initial vehicle velocity !!
• Needed final velocity !!
• Delta-V
�v2 =�
v22 + v2
a � 2v2va cos �i2
va =�
µ
r2
�2r1
r1 + r2
v2 =�
µ
r2
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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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Calculating Time in Orbit
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Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design
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Time in Orbit
• Period of an orbit
!• Mean motion (average angular velocity)
!• Time since pericenter passage
!➥M=mean anomaly
P = 2⇥
�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)
tan�
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= ⇥
�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
tan�
2=
�1 + e
1� etan
E
2=⇥ � = 160 deg
r = a(1� e cosE) = 12, 387 km
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
h⃗ = r⃗ × v⃗
p = a(1 − e2)
r =p
1 + e cos θ
rp = a(1 − e) ra = a(1 + e)
h =
√
µp
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Velocity Components in Orbit
34
r =p
1 + e cos �
vr =dr
dt=
d
dt
�p
1 + e cos �
⇥=�p(�e sin � d�
dt )(1 + e cos �)2
vr =pe sin �
(1 + e cos �)2d�
dt
1 + e cos � =p
r� vr =
r2 d�dt e sin �
p�⇤h = �⇤r ⇥�⇤v
Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design
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Velocity Components in Orbit (cont.)
35
~h = ~r ⇥ ~v h = rv cos � = r
✓rd✓
dt
◆= r2
d✓
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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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
v =
!
v2
h+
2µ
r
∆v =
!
v2
h+
2µ
r−
!
µ
r
v2
2� µ
r= � µ
2a=
v2h
2
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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)
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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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
40
vpm =
rv2h +
2µm
rLLO=
r0.8842 +
2(4667.9)
1878= 2.398
km
sec
�v = vpm � vcm = 2.398�r
4667.9
1878= 0.822
km
sec
Red text is a correction to original notes
Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design
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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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Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design
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Interplanetary Trajectory Types
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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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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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Interplanetary Delta-V
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C3 = V 2h
Hyperbolic excess velocity ⌘ Vh
Vreq =p
V 2esc + C3
�V =pV 2esc + C3 � Vc
2033 Window: �V = 3.55 km/sec
2037 Window: �V = 3.859 km/sec
�V in departure from 300 km LEO
Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design
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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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Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design
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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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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