Orbital Mechanics - University Of MarylandOrbital Mechanics ENAE 791 - Launch and Entry Vehicle...
Transcript of Orbital Mechanics - University Of MarylandOrbital Mechanics ENAE 791 - Launch and Entry Vehicle...
Orbital Mechanics ENAE 791 - Launch and Entry Vehicle Design
U N I V E R S I T Y O FMARYLAND
Orbital Mechanics• 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
1
© 2020 David L. Akin - All rights reserved http://spacecraft.ssl.umd.edu
Orbital Mechanics ENAE 791 - Launch and Entry Vehicle Design
U N I V E R S I T Y O FMARYLAND
Orbital Mechanics: 500 years in 40 min.• Newton’s Law of Universal Gravitation
• Newton’s First Law meets vector algebra
F =Gm1m2
r2
�⇥F = m�⇥a
2
Orbital Mechanics ENAE 791 - Launch and Entry Vehicle Design
U N I V E R S I T Y O FMARYLAND
Relative Motion Between Two Bodies
⇥F12 = force due to body 1 on body 2
⇥F12
⇥F21
⇥r1
⇥r2
m1
m2
3
m1d2 r1
dt2= G
m1m2
r212
r21
r21
= Gm1m2
r213 r21 = G
m1m2
r213 ( r2 − r1)
m2d2 r2
dt2= G
m1m2
r123 ( r1 − r2)
Orbital Mechanics ENAE 791 - Launch and Entry Vehicle Design
U N I V E R S I T Y O FMARYLAND
Gravitational Motion
“Equation of Orbit” - Orbital motion is simple harmonic motion
4
Let µ = G(m1 +m2)
d2~r
dt2+ µ
~r
r3= ~0
Orbital Mechanics ENAE 791 - Launch and Entry Vehicle Design
U N I V E R S I T Y O FMARYLAND
Orbital Angular Momentum
h is angular momentum vector (constant) =⇒r and v are in a constant plane
5
Orbital Mechanics ENAE 791 - Launch and Entry Vehicle Design
U N I V E R S I T Y O FMARYLAND
Fun and Games with Algebra
0
6
Orbital Mechanics ENAE 791 - Launch and Entry Vehicle Design
U N I V E R S I T Y O FMARYLAND
More Algebra, More Fun
7
Orbital Mechanics ENAE 791 - Launch and Entry Vehicle Design
U N I V E R S I T Y O FMARYLAND
Orientation of the Orbit
8
~e ⌘ eccentricity vector, in orbital plane
~e points in the direction of periapsis
Orbital Mechanics ENAE 791 - Launch and Entry Vehicle Design
U N I V E R S I T Y O FMARYLAND
Position in Orbit
θ = true anomaly: angular travel from perigee passage
9
Orbital Mechanics ENAE 791 - Launch and Entry Vehicle Design
U N I V E R S I T Y O FMARYLAND
Relating Velocity and Orbital Elements
10
Orbital Mechanics ENAE 791 - Launch and Entry Vehicle Design
U N I V E R S I T Y O FMARYLAND
Vis-Viva Equation
11
p ⌘ a(1� e2) =1� e2
2r � v2
µ
a =
✓2
r� v2
µ
◆�1
v2 = µ
✓2
r� 1
a
◆
v2
2� µ
r= � µ
2a
<--Vis-Viva Equation
Orbital Mechanics ENAE 791 - Launch and Entry Vehicle Design
U N I V E R S I T Y O FMARYLAND
Energy in Orbit• Kinetic Energy
• Potential Energy
• Total Energy
K.E. = 12mν 2 ⇒ K.E.
m=v2
2
P.E. = −mµr⇒
P.E.m
= −µr
Const. = v2
2−µr= −
µ2a
<--Vis-Viva Equation
12
Orbital Mechanics ENAE 791 - Launch and Entry Vehicle Design
U N I V E R S I T Y O FMARYLAND
Suborbital Tourism - Spaceship Two
13
Orbital Mechanics ENAE 791 - Launch and Entry Vehicle Design
U N I V E R S I T Y O FMARYLAND
How Close are we to Space Tourism?• Energy for 100 km vertical climb
• Energy for 200 km circular orbit
• Energy difference is a factor of 33!
� µ
rE + 100 km+
µ
rE= 0.965
km2
sec2= 0.965
MJ
kg
� µ
2(rE + 200 km)+
µ
rE= 32.2
km2
sec2= 32.2
MJ
kg
14
Orbital Mechanics ENAE 791 - Launch and Entry Vehicle Design
U N I V E R S I T Y O FMARYLAND
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
15
Orbital Mechanics ENAE 791 - Launch and Entry Vehicle Design
U N I V E R S I T Y O FMARYLAND
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
16
Orbital Mechanics ENAE 791 - Launch and Entry Vehicle Design
U N I V E R S I T Y O FMARYLAND
Classical Parameters of Elliptical Orbits
17
Orbital Mechanics ENAE 791 - Launch and Entry Vehicle Design
U N I V E R S I T Y O FMARYLAND
Basic Orbital Parameters• Semi-latus rectum (or parameter)
• Radial distance as function of orbital position
• Periapse and apoapse distances
• Angular momentumh = r × v
p = a(1 − e2)
r =p
1 + e cos θ
rp = a(1 − e) ra = a(1 + e)
h =√
µp
18
Orbital Mechanics ENAE 791 - Launch and Entry Vehicle Design
U N I V E R S I T Y O FMARYLAND
The Classical Orbital Elements
Ref: J. E. Prussing and B. A. Conway, Orbital Mechanics Oxford University Press, 1993
19
Orbital Mechanics ENAE 791 - Launch and Entry Vehicle Design
U N I V E R S I T Y O FMARYLAND
The Hohmann Transfer
vperigee
v1
vapogee
v2r1
r2
20
Orbital Mechanics ENAE 791 - Launch and Entry Vehicle Design
U N I V E R S I T Y O FMARYLAND
First Maneuver Velocities• Initial vehicle velocity
• Needed final velocity
• Required ΔV
v1 =
!
µ
r1
vperigee =
!
µ
r1
!
2r2
r1 + r2
∆v1 =
!
µ
r1
"!
2r2
r1 + r2
− 1
#
21
Orbital Mechanics ENAE 791 - Launch and Entry Vehicle Design
U N I V E R S I T Y O FMARYLAND
Second Maneuver Velocities
• Initial vehicle velocity
• Needed final velocity
• Required ΔV
∆v2 =
!
µ
r2
"
1 −
!
2r1
r1 + r2
#
vapogee =
!
µ
r2
!
2r1
r1 + r2
v2 =
!
µ
r2
22
Orbital Mechanics ENAE 791 - Launch and Entry Vehicle Design
U N I V E R S I T Y O FMARYLAND
Limitations on Launch Inclinations
Equator
23
Orbital Mechanics ENAE 791 - Launch and Entry Vehicle Design
U N I V E R S I T Y O FMARYLAND
Differences in Inclination
Line of Nodes
24
Orbital Mechanics ENAE 791 - Launch and Entry Vehicle Design
U N I V E R S I T Y O FMARYLAND
Choosing the Wrong Line of Apsides
25
Orbital Mechanics ENAE 791 - Launch and Entry Vehicle Design
U N I V E R S I T Y O FMARYLAND
Simple Plane Change
vperigee
v1vapogee
v2
Δv2
26
Orbital Mechanics ENAE 791 - Launch and Entry Vehicle Design
U N I V E R S I T Y O FMARYLAND
Optimal Plane Change
vperigee v1 vapogeev2
Δv2Δv1
27
Orbital Mechanics ENAE 791 - Launch and Entry Vehicle Design
U N I V E R S I T Y O FMARYLAND
First Maneuver with Plane Change Δi1
• Initial vehicle velocity
• Needed final velocity
• Required ΔV
v1 =�
µ
r1
vp =�
µ
r1
�2r2
r1 + r2
�v1 =�
v21 + v2
p � 2v1vp cos �i1
28
Orbital Mechanics ENAE 791 - Launch and Entry Vehicle Design
U N I V E R S I T Y O FMARYLAND
Second Maneuver with Plane Change Δi2
• Initial vehicle velocity
• Needed final velocity
• Required ΔV
�v2 =�
v22 + v2
a � 2v2va cos �i2
va =�
µ
r2
�2r1
r1 + r2
v2 =�
µ
r2
29
Orbital Mechanics ENAE 791 - Launch and Entry Vehicle Design
U N I V E R S I T Y O FMARYLAND
Sample Plane Change Maneuver
Optimum initial plane change = 2.20°
30
Orbital Mechanics ENAE 791 - Launch and Entry Vehicle Design
U N I V E R S I T Y O FMARYLAND
Calculating Time in Orbit
31
Orbital Mechanics ENAE 791 - Launch and Entry Vehicle 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
P = 2⇥
�a3
µ
n =�
µ
a3
M = nt = E � e sinE
32
Orbital Mechanics ENAE 791 - Launch and Entry Vehicle Design
U N I V E R S I T Y O FMARYLAND
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
33
Orbital Mechanics ENAE 791 - Launch and Entry Vehicle Design
U N I V E R S I T Y O FMARYLAND
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
34
Orbital Mechanics ENAE 791 - Launch and Entry Vehicle Design
U N I V E R S I T Y O FMARYLAND
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 sinEj
n =�
µ
a3= 1.650x10�4 rad
sec
e = 1� rp
a= 0.7270
35
Orbital Mechanics ENAE 791 - Launch and Entry Vehicle Design
U N I V E R S I T Y O FMARYLAND
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°
36
E = 2.317
tan�
2=
�1 + e
1� etan
E
2=⇥ � = 160 deg
r = a(1� e cosE) = 12, 387 km
Orbital Mechanics ENAE 791 - Launch and Entry Vehicle Design
U N I V E R S I T Y O FMARYLAND
Velocity Components in Orbit
37
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 791 - Launch and Entry Vehicle Design
U N I V E R S I T Y O FMARYLAND
Velocity Components in Orbit (cont.)
38
Orbital Mechanics ENAE 791 - Launch and Entry Vehicle Design
U N I V E R S I T Y O FMARYLAND
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.
39
Orbital Mechanics ENAE 791 - Launch and Entry Vehicle Design
U N I V E R S I T Y O FMARYLAND
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”
40
Orbital Mechanics ENAE 791 - Launch and Entry Vehicle Design
U N I V E R S I T Y O FMARYLAND
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 =
!
v2h
+2µ
r
∆v =
!
v2
h+
2µ
r−
!
µ
r
v2
2� µ
r= � µ
2a=
v2h
2
41
Orbital Mechanics ENAE 791 - Launch and Entry Vehicle Design
U N I V E R S I T Y O FMARYLAND
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
42
Orbital Mechanics ENAE 791 - Launch and Entry Vehicle Design
U N I V E R S I T Y O FMARYLAND
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
43
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
Orbital Mechanics ENAE 791 - Launch and Entry Vehicle Design
U N I V E R S I T Y O FMARYLAND
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
44
Orbital Mechanics ENAE 791 - Launch and Entry Vehicle Design
U N I V E R S I T Y O FMARYLAND
LOI ΔV Based on Landing Site
45
Orbital Mechanics ENAE 791 - Launch and Entry Vehicle Design
U N I V E R S I T Y O FMARYLAND
LOI ΔV Including Loiter Effects
46
Orbital Mechanics ENAE 791 - Launch and Entry Vehicle Design
U N I V E R S I T Y O FMARYLAND
Interplanetary Trajectory Types
47
Orbital Mechanics ENAE 791 - Launch and Entry Vehicle Design
U N I V E R S I T Y O FMARYLAND
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
48
Orbital Mechanics ENAE 791 - Launch and Entry Vehicle Design
U N I V E R S I T Y O FMARYLAND
C3 for Earth-Mars Transfer 1990-2045
49
Orbital Mechanics ENAE 791 - Launch and Entry Vehicle Design
U N I V E R S I T Y O FMARYLAND
Earth-Mars Transfer 2033
50
Orbital Mechanics ENAE 791 - Launch and Entry Vehicle Design
U N I V E R S I T Y O FMARYLAND
Earth-Mars Transfer 2037
51
Orbital Mechanics ENAE 791 - Launch and Entry Vehicle Design
U N I V E R S I T Y O FMARYLAND
Interplanetary Delta-V
52
C3 = V 2h
Hyperbolic excess velocity ⌘ Vh
Vreq =pV 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 791 - Launch and Entry Vehicle Design
U N I V E R S I T Y O FMARYLAND
Earth-Moon Potential Field
53
Orbital Mechanics ENAE 791 - Launch and Entry Vehicle Design
U N I V E R S I T Y O FMARYLAND
The Earth-Moon System
Earth
Moon
L1 L2L3
L4
L5
Note: Earth and Moon are in scale with size of orbits
Geostationary Orbit
Photograph of Earth and Moon taken by Mars Odyssey April 19, 2001 from a distance of 3,564,000 km
54
Orbital Mechanics ENAE 791 - Launch and Entry Vehicle Design
U N I V E R S I T Y O FMARYLAND
Free-Body Diagram with Spherical Planet
55
r
v
✓
� !
g
Orbital Mechanics ENAE 791 - Launch and Entry Vehicle Design
U N I V E R S I T Y O FMARYLAND
Orbital Planar State Equations
56
! = � � ✓
r
v
✓
� !
�g cos � = !v
�g sin � = v
g
Inertial angular velocity
Sum of accelerations normal to velocity vector
Sum of accelerations perpendicular to velocity vector
Orbital Mechanics ENAE 791 - Launch and Entry Vehicle Design
U N I V E R S I T Y O FMARYLAND
Orbital Planar State Equations (2)
57
r ˙✓ = v cos �
! = � � ˙✓ = � � v
rcos �
r = v sin �
�g cos � =
⇣� � v
rcos �
⌘v
�✓g � v2
r
◆cos � = �v
�✓1� v2
rg
◆g cos � = �v
Orbital Mechanics ENAE 791 - Launch and Entry Vehicle Design
U N I V E R S I T Y O FMARYLAND
Canonical Orbital Planar State Equations
58
Coupled first-order ODEs
� = �1
v
✓1� v2
v2c
◆g cos �
v = �g sin �
˙✓ =
v
rcos �
r = v sin �
g = go
⇣ro
r
⌘2
Orbital Mechanics ENAE 791 - Launch and Entry Vehicle Design
U N I V E R S I T Y O FMARYLAND
Numerical Integration - 4th Order R-K
59
˙y = f(t, x)
Given a series of equations
k1 = �t f (tn, yn)
k2 = �t f
✓tn +
�t
2, yn +
k12
◆
k3 = �t f
✓tn +
�t
2, yn +
k22
◆
k4 = �t f�tn +�t, yn + k3
�
¯yn+1 = yn +k16
+k23
+k33
+k46
+O(�t5)
Orbital Mechanics ENAE 791 - Launch and Entry Vehicle Design
U N I V E R S I T Y O FMARYLAND
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
60