Understanding Orbital Mechanics Through a Step-by-Step ... - AGI

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Understanding Orbital Mechanics Through a Step-by-Step
Examination of the Space-Based Infrared System (SBIRS)
Denny Sissom –Elmco, Inc.
May 2003
www.stk.com
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SSMD-1102-366 [1]

The Ground-Based Midcourse
Defense Architecture (2004)
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SSMD-0403-433 [2]
• Radars
• IFICS (In-Flight Interceptor Communications System)
• Ground-Based Interceptors
• Battle Management (BMC3)
• Space-Based Infrared System (SBIRS)
– SBIRS High GEO (Geo-Stationary Orbits)
– SBIRS High HEO (Highly-Elliptical Orbits)
– SBIRS Low (Low-Altitude Orbits)
– SBIRS Ground Station Processing (MCS)
www.stk.com
Pg 2 of 27
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SBIRS Model Overview
SBIRS High
Launch Detection
Boost Tracking
SBIRS Communication
Mission
Control
Station
(MCS)
Launch Detection
Boost Tracking
DSP/GEO
2D Detection
Report
Mission Control Station
•• One One Central CONUS Location
•Boost and and Coast Tracking
•Booster Typing
•Launch Point Estimation
•Impact Point Prediction
SBIRS Architecture
– Four Satellites in Geostationary
Orbits (GEO)
– Two Satellites in Highly
Elliptical Orbits (HEO)
– Twenty or more
Satellites in Low Earth
Orbit (LEO)
– Ground-Based Mission
Control Station (MCS)
SBIRS Low
Launch Detection
Boost Tracking
Mid-Course
Tracking
Discrimination
DSP Payload
• Scanner Only
-SWIR Band
-Periodic Revisit
• GEO Satellites
• Rotating Platform
• Provides 2D
Detection Reports to
MCS
LEO Payload
• Acquisition Sensor
-Wide FOV (WFOV)
-SWIR Band
-Boost Detection
• Track Sensor
-Narrow FOV
(NFOV)
-Multiple Wavebands
-2-Axis Gimbal
Control
-Precise Midcourse
Acquisition,
Tracking, &
Discrimination
GEO
Payload
• Scanner
–Rapid Global
Coverage
–SWIR, MWIR
Bands
–Taskable Scan
Rate and Revisit
• Starer
–SWIR, MWIR
Bands
–Taskable Revisit
• Follow-on and
replacement for
DSP
SSMD-0403-433 [3]
HEO Payload
• Highly Elliptical
Orbit (HEO)
• Scanner Only
- SWIR, MWIR
Bands
- Taskable Scan
Rate and Revisit
www.stk.com
Pg 3 of 27
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SBIRS Concept of Operations
Animation Showing Concept of Operations
From www.stk.com
• SBIRS High (GEO and/or HEO)
Acquire Target (SBIRS Low Can
Also Acquire Target)
• Data Transmitted From SBIRS
High To Mission Control Station
(MCS)
• Track Data Is Transmitted From
MCS To SBIRS Low
• SBIRS Low Acquires And Hands
Data Over From Acquisition
Sensor To Track Sensor
• Data Handed Over To Other SBIRS
Low Spacecraft and MCS
• Track Data Sent From
MCS To Battle Manager
SSMD-0403-433 [4]
www.stk.com
Pg 4 of 27
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Kepler’s Laws
SSMD-0403-433 [5]
Area 1 = Area 2
Planetary
Motion over
30 Days
Area 1
Area 2
Planetary
Motion over
30 Days
Average Distance
• Kepler’s First Law: The Orbits of Planets (or Satellites) are Ellipses with the Sun at a Focus
• Kepler’s Second Law: The Orbits of the Planets Sweep Out Equal Areas in Equal Time
• Kepler’s Third Law: The Square of the Orbit Period (The Time it Takes to Go Around Once)
is Proportional to the Cube of the Average Distance to the Sun
P
=
2π
www.stk.com
a
μ
3
Where:
P = Period (sec)
a = Semi-Major Axis (km)
m= Gravitational Parameter (km 3 /s 2 ) = GM earth
G = Universal Gravitational Constant (Nm 2 /kg 2 )
M earth
= Mass of the Earth (kg)
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Newton’s Law and the Restricted Two-
Body Equation of Motion
F
v
ma
Newton’s Second Law
SSMD-0403-433 [6]
v
=
2
−
µ
R
E
2
m
F g
=
Gm m
1
R
r − µ
E
F m
g
=
2
R
v
R
R
v
= ma
2
v
R
R
= mR
& v
v
R
R
& v µ
+ = 0
2
R R
Newton’s Law of Universal Gravitation
Newton’s Law of Universal Gravitation in
Vector Form with Earth as Central Body
(m E = GM earth = 3.986 x10 14 m 3 /s 2 )
Combining Newton’s Two Laws, assuming:
(1) No perturbations (drag, earth’s oblateness, other planets, etc.)
(2) Bodies are spherically symmetric
(3) m 1 >> m 2
We Get the Restricted Two-Body Equation of
Motion Which is a Second-Order, Non-Linear,
Vector Differential Equation – YUK!
This Equation Represents a Conic Section (Circle, Ellipse, Parabola, or Hyperbola)
www.stk.com
Pg 6 of 27
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A Few More Useful Equations for
Orbital Mechanics
E
=
v
H
1
2
v v
R × mV
v v v
h = R × V
= Angular Momentum
mV
2
mµ
−
R
Specific Angular Momentum, where
v
h ≡
Total Mechanical Energy for Orbiting Spacecraft
(Must remain constant!)
Apogee:
High PE = -mµ/R
Low KE = ½ mV 2
E
Earth
v
H
m
Perigee:
Low PE = -mµ/R
High KE = ½ mV 2
SSMD-0403-433 [7]
ε
V 2
= −
µ
2 R
ε
=
−
µ
2a
Specific Mechanical Energy, where
Shows We can Easily Find Specific Mechanical Energy Just
Knowing the Semi-Major Axis
- e is negative for circles and ellipses
- e is zero for parabolas
- e is positive for hyperbolas
ε
≡
E
m
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Pg 7 of 27
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Geocentric – Equatorial
Coordinate System
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SSMD-0403-433 [8]
• Origin –Center of Earth
• Fundamental Plane –Earth’s Equator
• Principle Direction (I-Axis)
– Vernal Equinox Direction Found by Drawing a Line from the Earth to the
Sun on the First Day of Spring
– Points at First Star in Aries Constellation (First Point of Aries)
– Denoted by Ram’s Head Symbol –
– Wanders Due to Earth Spin-Axis Wobble
– Because of the Wobble, Sometimes the Vernal Equinox Direction is
Specified at a Certain Time or “Epoch”
– Fixed at Vernal Equinox direction at Noon on January 1, 2000 at
Greenwich Meridian by International Astronomical Union (More Truly
Inertial)
• K-Axis
– North Pole
www.stk.com
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Semi-Major Axis and Eccentricity
The Size and Shape of a Orbit
SSMD-0403-433 [9]
e > 1
Semi-Major Axis
e = 1
Apogee radius
Perigee radius
Apogee Altitude
Perigee Altitude
Apogee
Center of
Ellipse
C
Perigee
0 < e < 1
e = 0
C = distance from center of Earth to center
of ellipse = eccentricity * semi major axis
ellipse
circle
• Size Determination: Semi-Major Axis
• Shape Determination: Eccentricity
www.stk.com
Pg 9 of 27
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Inclination
The Orientation of an Orbit
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• Tilt of Orbital Plane with Respect to Fundamental Plane (of Geocentric-
Equatorial Coordinate System)
• Angle Between Specific Angular Momentum Vector ( h = R × V ) and the
Vector Perpendicular to the Fundamental Plane Pointing Through the
North Pole (K-axis)
Inclination
Orbital Type
Diagram
• Ranges from 0° to 180°
Î
ĥ
i
Kˆ
Ĵ
0 or 180
90
0 £i < 90
90 < i £ 180
Equatorial
Polar
Direct or Prograde (Moves
in the Direction of Earth’s
Rotation)
Indirect or Retrograde
(Moves Against the
Direction of Earth’s
Rotation)
v
v
v
i =
90°
Ascending
node
Ascending
node
SSMD-0403-433 [10]
www.stk.com
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Right Ascension of Ascending Node (RAAN or Ω)
The “Swivel” of an Orbit
• Angle, Along the Equator, Between Principle Direction (i.e., First Point
of Aries) and the Point Where the Orbital Plane Crosses the Equator,
from South to North (The Ascending Node), Measured Eastward
• Not the Same As the Longitude of the Ascending Node
– RAAN Relative to Inertial Frame (Geocentric-Equatorial)
– Longitude of Ascending Node Relative to Rotating Earth
• Ranges from 0° to 360°
Kˆ
SSMD-0403-433 [11]
Î
Equatorial
Plane
Ω
Ĵ
Ascending
Node
www.stk.com
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Argument of Perigee (ω)
The Orientation of the Orbit within the Orbital Plane
SSMD-0403-433 [12]
• Angle Along Orbital Path Between the Ascending Node and the Perigee
• Always measured Along the Orbital Path in Direction of Spacecraft
Motion
• Perigee – Closest Approach to Earth
• Ranges from 0° to 360°
Kˆ
Perigee
ω
Ĵ
Î
www.stk.com
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True Anomaly at Epoch
The Spacecraft’s Location within an Orbit
SSMD-0403-433 [13]
• Angle Along Orbital Path from Perigee to Spacecraft’s Position
• Always Measured Along Orbital Path in Direction of Spacecraft Motion
• The Only Orbital Element Set Parameter That Varies with Time as the
Spacecraft Travels Around its Fixed Orbit, Assuming a Spherically-
Symmetric Earth (A So-So Assumption)
Vˆ
Rˆ
ν
Perigee
www.stk.com
Pg 13 of 27
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Summary of Orbital Elements
SSMD-0403-433 [14]
Element
Name
Description
Range of Values
Undefined
a
Semimajor Axis
Size
Depends on the
Conic Section
Never
e
Eccentricity
Shape
e = 0: Circle
0 < e < 1: ellipse
Never
i
Inclination
Tilt, angle from Kˆ unit
vector to specific
angular momentum
vector ĥ
0 £i £ 180
Never
W
Right ascension
of the ascending
node
Swivel, angle from
vernal equinox to
ascending node
0 £W£360
When i = 0 or 180
(equatorial orbit)
w
Argument of
perigee
Angle from ascending
node to perigee
0 £w£360
When i = 0 or 180
(equatorial orbit) or e = 0
(circular orbit)
n
True anomaly
Angle from perigee to
the spacecraft’s position
0 £n£360
When e = 0 (circular orbit)
www.stk.com
Pg 14 of 27
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Alternate Orbital Elements
Element
u
• A Circular Orbit
– No Argument of Perigee
– No True Anomaly
• An Equatorial Orbit
– No RAAN
– No Argument of Perigee
Name
Argument of
latitude
What Do We Do With:
Description
Angle from ascending node
to the spacecraft’s position
• A Circular Equatorial Orbit
– No RAAN
– No Argument of Perigee
– No True Anomaly
Range of Values
0 £u£360
Undefined
SSMD-0403-433 [15]
Use when there is no perigee (e =
0)
P
Longitude of
perigee
Angle from the principal
direction to perigee
0 £P£360
Use when equatorial (i = 0 or
180 ) because there is no
ascending node
l
True longitude
Angle from the principal
direction to the spacecraft’s
position
0 £l£360
Use when there is no perigee and
ascending node (e = 0 and i = 0
or 180 )
www.stk.com
Pg 15 of 27
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SBIRS High Scenario
SSMD-0403-433 [16]
• SBIRS High is a “Molniya” Type Orbit
• Russian word for “Zipper” or “Lightning”
• Large Dwell Time over Northern Hemisphere
• Usually a 12-Hour Orbit with High Eccentricity (0.7)
and Perigee in Southern Hemisphere
• Has Inclination of 63.4° (No Rotation of Perigee)
• Covers High Latitudes and Polar Regions Very Well
www.stk.com
Pg 16 of 27
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SBIRS Low Coverage Studies
SSMD-0403-433 [17]
SBIRS Low Constellation Showing Threat Object Coverage
(Sensor Footprints in Green, Sensor Acquisitions in Yellow)
• SBIRS Low Constellation As Implemented In TESS
• Coverage Almost Complete Utilizing 24 Satellites
• Orbital Element Set Propagation Within TESS
www.stk.com
Pg 17 of 27
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SBIRS DSP (GEO)
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SSMD-0403-433 [18]
From www.stk.com
• Geostationary Orbits (Fixed ECR)
• Above and Below-the-Horizon Viewing Ability
www.stk.com
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In Summary
• Excellent References
– Expensive: Understanding Space – An Introduction to Astronautics, Jerry
Jon Sellers
$66.00 at www.walmart.com
– Cheap: Fundamentals of Astrodynamics, Roger R. Bate
$9.00 at www.walmart.com
Introduction to Space Dynamics, William Tyrrell Thomson
$9.00 at www.walmart.com
– Free: TRW Space Data, Neville J. Barter, editor
Free from TRW Space and Electronics Group
• Excellent Web Site
– www.heavens-above.com
– Iridium Flares, ISS, HST, etc.
• Excellent Software
– Satellite Tool Kit from Analytical Graphics, Inc. (www.stk.com)
– Price: Free to Over $100,000
• Training Available for Basic Orbital Mechanics
www.stk.com
Pg 19 of 27
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SSMD-0403-433 [19]

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SSMD-0403-433 [20]
Supplemental Charts
www.stk.com
Pg 20 of 27
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Ground-Based Midcourse
Defense Architecture (2004)
SSMD-0403-433 [21]
GBIs
IFICS
BMC3
BMC3
Cobra Dane
IFICS
GBIs
IFICS
UEWR
GBIs
IFICS
BMC3
GBR-P
AEGIS
GBIs
SBIRS MCS
IFICS
www.stk.com
Pg 21 of 27
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GMD with SBIRS High and DSP
SSMD-0403-433 [22]
From www.stk.com
www.stk.com
Pg 22 of 27
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SBIRS Waveband Utilization
• SBIRS DSP, High, and Low
Utilize Different Sensor
Wavebands
SBIRS High
• MWIR (3-8 mm)
• SWIR (1-3 mm)
DSP/GEO
• SWIR (1-3 mm)
SSMD-0403-433 [23]
• Different Target Types are Visible
in Different Wavelengths
• Synergy Between Satellites Allow
Full Tracking of Threat Objects
from Initial Launch Through Mid-
Course
PBVs
PBV
Plumes
Upper
Stage
Boost
Phase
Low-
Altitude
Boost
Phase
• Provides Extended Capability for
Strategic and Theater Missile
Defense
SBIRS Low
• LWIR (8-14 mm)
• MWIR (3-8 mm)
• SWIR (1-3 mm)
• Visible (0.4-0.7 mm)
Mid-
Course
Tracking
Visible Near Infrared Middle Infrared Far Infrared Extreme Infrared
V B G Y OR
0.4
0.6
0.8
1
1.5
2
3
4
6
8
10
15
20
30
www.stk.com
Pg 23 of 27
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Effects of Earth’s Oblateness
on Orbiting Spacecraft
22 km
Rˆ
F v
J 2
Nodal Regression Rate
SSMD-0403-433 [24]
22 km
Perigee Rotation Rate
.
• Equatorial Bulge Causes Slight Shift in Direction
Gravity Pulls Spacecraft
• Modeled by Complex Mathematics Referred to as
the “J2 Effect”
• Earth is 22 km Bigger (radius) at Equator
• Causes Nodal Regression Rate (Movement of the
RAAN, Ω) and . a Perigee Rotation Rate (ω)
www.stk.com
Graphs from “Understanding Space” by Jerry Jon Sellers
Pg 24 of 27
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Sun Synchronous Orbits
If Someone Gives You Lemons, Make Lemonade! (Part 1)
SSMD-0403-433 [25]
• Despite the Complexities That the “J2 Effect” Cause, There are Advantages
• Sun-Synchronous Orbits Take Advantage of the Rate of Change of the RAAN
• Inclination is Set to Give Approximately a One-Degree Nodal Regression Eastward per day (Note that the
Earth Moves 0.9863 Degrees per day in its Orbit Around the Sun (i.e., 360 /365 days)
• Spacecraft’s Orbital Plane Always Maintains Same Orientation to Sun
– Spacecraft Always Sees Same Sun Angle When It Passes Over a Particular Point on Earth
– Sun’s Shadows Cast by Objects on Earth’s Surface Will Not Change When Pictures are Taken Days or Weeks Apart
– Good for Remote Sensing, Reconnaissance, Weather, etc.
Earth moves
around the Sun at
1° /day
Orbital plane
rotates at ~1° /day
due to earth’s
oblateness
Inclination = 97.03
Orbital plane
Sun line
Sun angle
www.stk.com
Pg 25 of 27
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Molniya Orbits
If Someone Gives You Lemons, Make Lemonade! (Part 2)
SSMD-0403-433 [26]
• Another Advantage of the “J2 Effect”
• Molniya –Russian word for “Zipper”
or “Lightning”
• Large Dwell Time over Northern
Hemisphere
• Usually a 12-Hour Orbit with High
Eccentricity (0.7) and Perigee in
Southern Hemisphere
• Has Inclination of 63.4 (No Rotation
of Perigee)
• Covers High Latitudes and Polar
Regions Very Well
www.stk.com
Pg 26 of 27
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Geosynchronous Orbit
No Perigee Rotation
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SSMD-0403-433 [27]
• Orbits Every 24 Hours
• Inclination of 63.4 degrees
• No Perigee Rotation
www.stk.com
Pg 27 of 27
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