AA 4362 Astrodynamics

1
AA 4362 Astrodynamics
Orbit Perturbations and Orbit Decay
Chapter 8,Sellers sections 8.2, 8.3 (pp. 272-283)
Week 9
2
The N-Body Problem
Chaotic, Unsolvable Computational problem
3
Generalized Three-Body Problem
"three body problem" is the solution of the
motion of three bodies under their mutual
attraction. General three-body motion has
chaotic properties. Even the general "restricted
three body problem" where one of the bodies is
very small--e.g. Earth, Moon and spacecraft--is
analytically Insoluble Solutions must be
generated using numerical simulation Specific
solutions exist, like the ones in which the
spacecraft is positioned at one of the Lagrange
points.
4
Restricted Three Body Problem
5
Restricted Three Body Problem
Ignore Mass of Space Craft
6
Acceleration of Target Relative to Earth
7
Acceleration of Target Relative to Earth (contd)
8
Acceleration of Target Relative to Sun
9
Acceleration of Target Relative to Sun (contd)
10
(No Transcript)
11
What are Other factors that Perturb the Orbit?

• Any disturbance in the regular motion of a
  satellite resulting from a force other than those
  causing regular motion
• Non-spherical earth (Oblateness)
• Sun/moon/planets gravity(Third bodies)
• Atmospheric drag

12
The Earth is Not a Perfect Sphere
In actuality, Earth is an Ellipsoid
With lots of warts on the surface!
(www.space.com)
Gravitational Potential Model of the Earth,
March 19, 2002

13
Mean Equatorial Bulge
14
Gravitational Potential Theory
Irregularities in the Earths Gravitational
Field modeled by a complex Mathematical Model
referred to as Potential Theory
15
Gradient Operator in Spherical Coordinates
16
Laplacian Operator in Spherical Coordinates
(contd)
17
Gravitational PotentialSpherical Earth
Energy Stored when an object of mass M, is pulled
to from a near earth distance r From infinity
F
18
Gravitational PotentialSpherical Earth (contd)
Gravity produces no Orbital torque for
spherical Earth Gravitational Force acts
Through center of earth
OK!
cos(?)
19
Gravitational Potential Spherical
Earth (contd)
Satisfies Laplaces equation
OK!
20
Gravitational PotentialNon-Spherical Earth
(contd)
For a more realistic Earth model warts and
all we have to expand The potential function to
include Additional gravitational terms
the Spherical harmonics

Spherical Harmonics
Ouch! This hurts
21
Gravitational PotentialNon-Spherical Earth
(contd)
Fortunately, for a planet that is symmetrical
about the Z (polar) axis, and has North/South
symmetry, the Form of the potential function
reduces Considerably The so-called
gravitational Multi-pole expansion
22
Gravitational PotentialNon-Spherical Earth
(contd)
P2n(cos(?) ) are the legendre polynomials,
and J2n are the gravitational moments
J-numbers Re Equatorial Planetary radius
f is the co-latitude (measured from polar axis)
23
Gravitational PotentialNon-Spherical Earth
(contd)
Legendre polynomials
24
Gravitational PotentialNon-Spherical Earth
(contd)
J2 is dominant 250 times bigger
25
Gravitational PotentialNon-Spherical Earth
(contd)
If the harmonics are truncated after second
order the resulting J2 model is
J2 is the second order gravitational moment
Re Equatorial Planetary radius f is the
co-latitude (measured from polar axis)
26
Gravitational Potential TheoryJ2 Model

27
Gravitational Potential TheoryJ2 Model (contd)
From harmonic analysis
r(r,m) -gt planetary density function
J2 Gravitational Moment is approximated by

28
Gravitational Potential TheoryJ2 Model (contd)
What is J2 for earth based on this model


0.00108144

29
Gravitational Potential TheoryJ2 Model (contd)
What is J2 for earth based on this model
Accepted values
0.00108144
Pretty Good Model 0.1 accurate

30
Force of gravity, J2 model
Prove that J2 gravity no longer acts thru
earth center
31
Force of gravity, J2 model(contd)
32
Force of gravity, J2 model(contd)
33
Force of gravity, J2 model(contd)
Gravitational acceleration vector does not pass
through center of earth
34
Force of gravity, J2 model(contd)
At Earth surface
Radial Acceleration of gravity
35
Force of gravity, J2 model(contd)
At Earth surface Radial Acceleration of gravity
Normalized to 1g
g/g0
North pole
l
South pole
equator
36
Force of gravity, J2 model(contd)
At Earth surface
North/South (longitudinal) Acceleration of Gravity
37
Force of gravity, J2 model(contd)
At Earth surface North/South (longitudinal)Accel
eration of gravity Normalized to 1g
g/g0
North pole
l
South pole
equator
38
Gravitational Potential Theory, J2 Model
(contd)
J1 --gt Gravitational Potential of Spherical
earth J2 --gt Accounts for Equatorial Bulge
So-named J2 Effect
J2 -- torques the orbit
39
How Does J2 Torque the Orbit? (contd)
Spherical Earth Gravitational force acts
On a straight line between CG Of satellite and
the center of the earth No moment arm to act
on .. Thus no torque
40
How Does J2 Torque the Orbit? (contd)
Oblate (J2) Earth Spacecraft is closer to
the centroid of one hemisphere than the
centroid of the other hemisphere
Distance from hemisphere centroid
41
How Does J2 Torque the Orbit? (contd)
Oblate (J2) Earth Gravitational force no
longer acts through center of the earth
Gravitational Force Vector
42
J2 Gravity Model (revisited)
Re-write in terms of in, ik
43
Rotation from l,d ton,k
Ascending orbit
44
Rotation from l,d ton,k
Descending orbit
45
Rotation from l,d ton,k(contd)
Ascending orbit
Descending Orbit
Why?
For J2 Model
46
J2 Gravity Model orbital plane effects
depending on ascending, descending
47
J2 Gravity Model orbital plane effects (contd)
depending on ascending, descending
48
J2 Gravity Model orbital plane effects (contd)
Effect of Fl
Effect of F?
49
How Does J2 Torque the Orbit? (contd)
Oblate (J2) Earth Net torque on orbit
Due to vector results in precession of the
line-of-nodes
50
Precession of the Right Ascension (J2 Nodal
Regression)
When Averaged Over an entire orbit
Nodal Precession Earth orbit
51
Precession of the Right Ascension (J2 Nodal
Regression)
General Formula
When Averaged Over an entire orbit
52
Precession of the Right Ascension (J2 Nodal
Regression)
Earth Formula
When Averaged Over an entire orbit
?/civil day
53
Precession of the Right Ascension(Nodal
Regression)
When Averaged Over an entire orbit
deg/ civil day
54
Sun-Synchronous Orbit

• Approx 98 degree Inclination
• Circular or Elliptical LEO
• Sun Angle is Always same (more later)
• Missions
• Electro Optical Imaging
• Weather (DMSP)

5 4 3 2 1
GROUND TRACE
55
Sun Synchronous Orbit
Takes Advantage of the fact that precession of
the line of nodes is the same rate as the earths
angular velocity around the sun (about 0.9856?
per day) (retrograde orbits)
Summer
Spring
Fall
Winter
56
Sun Synchronous Orbit (concluded)
57
J2 Effect Perigee Rotation

• Orbit rotates in orbital plane

58
J2 Effect Perigee Rotation
Asymmetrically distributed torque forces
perigee rotation within Orbital plane
59
Variation of Argument of Perigee
When Averaged Over an entire orbit
60
Variation of Argument of Perigee(contd)
When Averaged Over an entire orbit
General Formula
61
Variation of Argument of Perigee(contd)
Earth Formula
?/ civil day
When Averaged Over an entire orbit
62
When Averaged Over an entire orbit
Variation of Argument of Perigee
civil
An Interesting result
Perigee does not rotate
63
Molniya Orbits
Molniya Orbits exploit this quirk of nature

• 63.4 /116.6 Degree Inclination
• 500 km perigee - 40,000 km apogee
• Missions Comm / Surveillance

64
(No Transcript)
65

66

67

68
General Precession

69
Atmospheric Drag
70
Effect of Drag on Orbital Energy
71
Orbital Energy Analysis
72
Orbital Energy Analysis (contd)
D
73
Orbital Energy Analysis (contd)
74
Orbital Energy Analysis (contd)
75
Orbital Energy Analysis (contd)
76
Orbital Energy Analysis (contd)
77
Aerodynamic Forces
78
Drag Force
g
79
Drag Force (contd)
80
Drag Force (contd)
Air sticks to Earth boundary
81
Drag Force (contd)
82
Bound the Velocity error
83

84

85

86
Orbit Decay Equation
87
Orbit Decay Equation (contd)
88
Ballistic Coefficient (Orbit decay parameter)
NORAD codes like SGP4 use a B coefficient
instead of Drag Coefficient More Direct
Measure of orbit Decay
kg/m3

89
Ballistic Coefficient (contd)
Just to make it more confusing .. NORAD
Two-line-element (TLE) sets display B In units
of earth radii -1
kg/m3
Drag force
90
Ballistic Coefficient (contd)
Drag force
91
Orbit decay equation
92
Orbit decay equation (contd)
1
1
93

94
Numerical Example Orbit Decay of Russian Zarya
Module WHAT IF ENGINE QUIT WORKING ?
95

Russian Zarya Module

96

Russian Zarya Module

97
Russian Zarya Moduke
Russian Zarya Module

98
A Little Altitude at LEO makes a big Difference
99
A Little Altitude at LEO makes a big Difference
(contd)
100
A Little Altitude at LEO makes a big Difference
(contd)
h