Understanding Orbital Mechanics Through A Step-by-step
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UNCLASSIFIED 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 Pg 1 of 27 UNCLASSIFIED SSMD-1102-366 [1] UNCLASSIFIED 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) www.stk.com Pg 2 of 27 UNCLASSIFIED UNCLASSIFIED 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) www.stk.com 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 UNCLASSIFIED 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 UNCLASSIFIED SBIRS Concept of Operations SSMD-0403-433 [4] • 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 Animation Showing Concept of Operations • Data Handed Over To Other SBIRS Low Spacecraft and MCS From www.stk.com • Track Data Sent From MCS To Battle Manager www.stk.com Pg 4 of 27 UNCLASSIFIED UNCLASSIFIED 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 a3 P = 2π μ www.stk.com Where: P = Period (sec) a = Semi-Major Axis (km) µ= Gravitational Parameter (km3/s2) = GMearth G = Universal Gravitational Constant (Nm2/kg2) Mearth = Mass of the Earth (kg) Pg 5 of 27 UNCLASSIFIED UNCLASSIFIED Newton’s Law and the Restricted TwoBody Equation of Motion Fg = SSMD-0403-433 [6] v v F = ma Newton’s Second Law Gm1m2 R2 Newton’s Law of Universal Gravitation v r − µEm R Fg = R2 R v v& − µE m R v & = ma = mR 2 R R v v&& µ R R+ 2 =0 R R Newton’s Law of Universal Gravitation in Vector Form with Earth as Central Body (µE = GMearth = 3.986 x1014 m3/s2) Combining Newton’s Two Laws, assuming: (1) (2) (3) No perturbations (drag, earth’s oblateness, other planets, etc.) Bodies are spherically symmetric m1 >> m2 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 UNCLASSIFIED UNCLASSIFIED A Few More Useful Equations for Orbital Mechanics v v v H = R × mV v v v h = R ×V E= 1 mµ mV 2 − 2 R SSMD-0403-433 [7] Angular Momentum v v H Specific Angular Momentum, where h ≡ m Total Mechanical Energy for Orbiting Spacecraft (Must remain constant!) Apogee: High PE = -mµ/R Low KE = ½ mV2 V2 µ ε= − 2 R µ ε =− 2a www.stk.com E Perigee: Low PE = -mµ/R High KE = ½ mV2 Earth Specific Mechanical Energy, where ε ≡ E m Shows We can Easily Find Specific Mechanical Energy Just Knowing the Semi-Major Axis - ε is negative for circles and ellipses - ε is zero for parabolas - ε is positive for hyperbolas Pg 7 of 27 UNCLASSIFIED UNCLASSIFIED 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 www.stk.com Pg 8 of 27 UNCLASSIFIED UNCLASSIFIED Semi-Major Axis and Eccentricity The Size and Shape of a Orbit SSMD-0403-433 [9] e>1 e=1 Semi-Major Axis Apogee radius Perigee radius Apogee Altitude Apogee Center of Ellipse Perigee Altitude Perigee C e=0 0<e<1 circle C = distance from center of Earth to center of ellipse = eccentricity * semi major axis ellipse • Size Determination: Semi-Major Axis • Shape Determination: Eccentricity www.stk.com Pg 9 of 27 UNCLASSIFIED UNCLASSIFIED Inclination The Orientation of an Orbit SSMD-0403-433 [10] • Tilt of Orbital Plane with Respect to Fundamental Plane (of GeocentricEquatorial Coordinate System) v v v • 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 0° or 180° Equatorial 90° Polar 0° ≤ i < 90° Direct or Prograde (Moves in the Direction of Earth’s Rotation) • Ranges from 0° to 180° ĥ i K̂ 90° < i ≤ 180° Ĵ Indirect or Retrograde (Moves Against the Direction of Earth’s Rotation) Iˆ www.stk.com Pg 10 of 27 UNCLASSIFIED i= 90° Ascending node Ascending node UNCLASSIFIED Right Ascension of Ascending Node (RAAN or Ω) The “Swivel” of an Orbit SSMD-0403-433 [11] • 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 K̂ • Ranges from 0° to 360° Ω Iˆ Equatorial Plane www.stk.com Pg 11 of 27 UNCLASSIFIED Ĵ Ascending Node UNCLASSIFIED 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 K̂ Perigee • Ranges from 0° to 360° ω Ĵ Iˆ www.stk.com Pg 12 of 27 UNCLASSIFIED UNCLASSIFIED 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 SphericallySymmetric Earth (A So-So Assumption) Vˆ ν R̂ www.stk.com Pg 13 of 27 UNCLASSIFIED Perigee UNCLASSIFIED 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 K̂ unit vector to specific angular momentum vector ĥ 0° ≤ i ≤ 180° Never Ω Right ascension of the ascending node Swivel, angle from vernal equinox to ascending node 0° ≤ Ω ≤ 360° When i = 0° or 180° (equatorial orbit) ω Argument of perigee Angle from ascending node to perigee 0° ≤ ω ≤ 360° When i = 0° or 180° (equatorial orbit) or e = 0° (circular orbit) ν True anomaly Angle from perigee to the spacecraft’s position 0° ≤ ν ≤ 360° When e = 0 (circular orbit) www.stk.com Pg 14 of 27 UNCLASSIFIED UNCLASSIFIED Alternate Orbital Elements SSMD-0403-433 [15] • A Circular Orbit? What Do We Do With: • A Circular Equatorial Orbit? – No Argument of Perigee – No True Anomaly – No RAAN – No Argument of Perigee – No True Anomaly • An Equatorial Orbit? – No RAAN – No Argument of Perigee Element Name Description Range of Values Undefined u Argument of latitude Angle from ascending node to the spacecraft’s position 0° ≤ u ≤ 360° Use when there is no perigee (e = 0) Π Longitude of perigee Angle from the principal direction to perigee 0° ≤ Π ≤ 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 UNCLASSIFIED UNCLASSIFIED 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 UNCLASSIFIED UNCLASSIFIED 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 UNCLASSIFIED UNCLASSIFIED SBIRS DSP (GEO) SSMD-0403-433 [18] From www.stk.com • Geostationary Orbits (Fixed ECR) • Above and Below-the-Horizon Viewing Ability www.stk.com Pg 18 of 27 UNCLASSIFIED UNCLASSIFIED In Summary SSMD-0403-433 [19] • Excellent References – Expensive: – Cheap: – Free: Understanding Space – An Introduction to Astronautics, Jerry Jon Sellers $66.00 at www.walmart.com 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 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 UNCLASSIFIED UNCLASSIFIED SSMD-0403-433 [20] Supplemental Charts www.stk.com Pg 20 of 27 UNCLASSIFIED UNCLASSIFIED Ground-Based Midcourse Defense Architecture (2004) SSMD-0403-433 [21] GBIs IFICS BMC3 Cobra Dane IFICS BMC3 GBIs IFICS UEWR GBIs IFICS BMC3 GBR-P SBIRS MCS AEGIS GBIs IFICS www.stk.com Pg 21 of 27 UNCLASSIFIED UNCLASSIFIED GMD with SBIRS High and DSP SSMD-0403-433 [22] From www.stk.com www.stk.com Pg 22 of 27 UNCLASSIFIED UNCLASSIFIED SBIRS Waveband Utilization SSMD-0403-433 [23] SBIRS High DSP/GEO • SBIRS DSP, High, and Low Utilize Different Sensor Wavebands • MWIR (3-8 µm) • SWIR (1-3 µm) • SWIR (1-3 µm) • Different Target Types are Visible in Different Wavelengths • Synergy Between Satellites Allow Full Tracking of Threat Objects from Initial Launch Through MidCourse • Provides Extended Capability for Strategic and Theater Missile Defense Visible Near Infrared Middle Infrared PBVs SBIRS Low • LWIR (8-14 µm) • MWIR (3-8 µm) • SWIR (1-3 µm) • Visible (0.4-0.7 µm) Far Infrared Extreme Infrared V B G Y OR 0.4 0.6 0.8 1 www.stk.com 1.5 2 3 4 6 8 10 PBV Plumes 15 20 30 Pg 23 of 27 UNCLASSIFIED MidCourse Tracking Upper Stage Boost Phase LowAltitude Boost Phase UNCLASSIFIED Effects of Earth’s Oblateness on Orbiting Spacecraft SSMD-0403-433 [24] 22 km Nodal NodalRegression RegressionRate Rate R̂ v FJ 2 22 km • • • • Perigee PerigeeRotation RotationRate 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 UNCLASSIFIED UNCLASSIFIED 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 www.stk.com Sun angle Pg 25 of 27 UNCLASSIFIED UNCLASSIFIED 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 UNCLASSIFIED UNCLASSIFIED Geosynchronous Orbit No Perigee Rotation SSMD-0403-433 [27] • Orbits Every 24 Hours • Inclination of 63.4 degrees • No Perigee Rotation www.stk.com Pg 27 of 27 UNCLASSIFIED
