Understanding Orbital Mechanics Through a Step-by-Step Examination of the Space-Based Infrared System (SBIRS)

Understanding Orbital Mechanics Through a Step-by-Step Examination of the Space-Based Infrared System (SBIRS) Denny Sissom Elmco, Inc. May 2003 Pg 1 of 27 SSMD-1102-366 [1]

The Ground-Based Midcourse Defense Architecture (2004) 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) Pg 2 of 27

SBIRS Model Overview SSMD-0403-433 [3] SBIRS Communication SBIRS High Launch Detection Boost Tracking Launch Detection Boost Tracking DSP/GEO Mission Control Station (MCS) Launch Detection Boost Tracking Mid-Course Tracking Discrimination 2D Detection Report Mission MissionControl ControlStation Station One Central CONUS One Central CONUSLocation Location Boost Boostand andcoast CoastTracking Tracking Booster BoosterTyping Typing Launch LaunchPoint PointEstimation Estimation Impact Point Prediction 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) LEO Payload SBIRS Low 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 DSP Payload Scanner Only - SWIR Band - Periodic Revisit GEO Satellites Rotating Platform Provides 2D Detection Reports to MCS Pg 3 of 27 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 HEO Payload Highly Elliptical Orbit (HEO) Scanner Only - SWIR, MWIR Bands - Taskable Scan Rate and Revisit

SBIRS Concept of Operations Animation Showing Concept of Operations From 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] Pg 4 of 27

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 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) Pg 5 of 27

Newton s Law and the Restricted Two- Body Equation of Motion v F = v ma Newton s Second Law SSMD-0403-433 [6] µ R E 2 m Gm1m F g = 2 R R F m v r µ E g = 2 R R v R v = ma = mr & v R v R R & v µ + = 0 2 R R 2 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) Pg 6 of 27

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 Pg 7 of 27

Geocentric Equatorial Coordinate System 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 Pg 8 of 27

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 Pg 9 of 27

Inclination The Orientation of an Orbit 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] Pg 10 of 27

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 Pg 11 of 27

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 Pg 12 of 27

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 Pg 13 of 27

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) Pg 14 of 27

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 ) Pg 15 of 27

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 Pg 16 of 27

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 Pg 17 of 27

SBIRS DSP (GEO) SSMD-0403-433 [18] From Geostationary Orbits (Fixed ECR) Above and Below-the-Horizon Viewing Ability Pg 18 of 27

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. () Price: Free to Over $100,000 Training Available for Basic Orbital Mechanics Pg 19 of 27 SSMD-0403-433 [19]

SSMD-0403-433 [20] Supplemental Charts Pg 20 of 27

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 Pg 21 of 27

GMD with SBIRS High and DSP SSMD-0403-433 [22] From Pg 22 of 27

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 Pg 23 of 27

Effects of Earth s Oblateness on Orbiting Spacecraft 22 km Rˆ Nodal Regression Rate SSMD-0403-433 [24] F v J 2 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) Graphs from Understanding Space by Jerry Jon Sellers Pg 24 of 27

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 Pg 25 of 27

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 Pg 26 of 27

Geosynchronous Orbit No Perigee Rotation SSMD-0403-433 [27] Orbits Every 24 Hours Inclination of 63.4 degrees No Perigee Rotation Pg 27 of 27