Colorado Springs Cadet Squadron Lt Col M. T. McNeely ORBITAL MECHANICS !! INTRO TO SPACE COURSE

ORBITAL MECHANICS Lesson 1 F Origins F Physical Laws F Requirements for Injection F Classifications of Orbits F Coordinate Reference Systems F Orbital Elements F Ground Tracks F Perturbations F Launch Considerations

ORIGINS Nicholas Copernicus F Revived Helio-centric model F Believed planetary orbits were circles

ORIGINS Tycho Brahe F Introduced precision into astronomical measurements F Mentor to Johannes Kepler

ORIGINS Johannes Kepler F Derived 3 laws based upon his observations of planetary motion

PHYSICAL LAWS Kepler’s 1st Law: Law of Ellipses The orbits of the planets are ellipses with the sun at one focus

PHYSICAL LAWS Ellipses FOCI Period (T) Semi-Major Axis (a) Semi-Minor Axis (b)

PHYSICAL LAWS

PHYSICAL LAWS Kepler’s 2nd Law: Law of Equal Areas The line joining the planet to the center of the sun sweeps out equal areas in equal times T6 T5 T4 T3 T2 T1 A2 A3A4 A5 A6 A1

PHYSICAL LAWS Kepler’s 2nd Law: Law of Equal Areas

t0t0 t3t3 t1t1 t2t2 Area 1 Area 2 t 1 -t 0 = t 3 -t 2 Area 1 = Area 2 Satellite travels at varying speeds

PHYSICAL LAWS Kepler’s 3rd Law: Law of Harmonics The squares of the periods of two planets’ orbits are proportional to each other as the cubes of their semi- major axes: T 1 2 /T 2 2 = a 1 3 /a 2 3 In English: Orbits with the same semi- major axis will have the same period

ORBITAL MECHANICS Lesson 1 F Origins F Physical Laws F Requirements for Injection F Classifications of Orbits F Coordinate Reference Systems F Orbital Elements F Ground Tracks F Perturbations F Launch Considerations

PHYSICAL LAWS Sir Isaac Newton F Derived three laws of motion F Derived the Law of Universal Gravitation F Explained why Kepler’s laws worked

PHYSICAL LAWS Newton’s 1st Law: Law of Inertia F Every body continues in a state of uniform motion unless it is compelled to change that state by a force imposed upon it

PHYSICAL LAWS Newton’s 2nd Law: Law of Momentum F Change in momentum is proportional to and in the direction of the force applied F Momentum equals mass x velocity F Change in momentum gives: F = ma F F

PHYSICAL LAWS Newton’s 3rd Law: Action - Reaction F For every action, there is an equal and opposite reaction F Hints at conservation of momentum

PHYSICAL LAWS Newton’s Law of Universal Gravitation Between any two objects there exists a force of attraction that is proportional to the product of their masses and inversely proportional to the square of the distance between them F g = G ( ) M1m2M1m2 D2D2

ORBITAL MECHANICS Lesson 1 F Origins F Physical Laws F Requirements for Injection F Classifications of Orbits F Coordinate Reference Systems F Orbital Elements F Ground Tracks F Perturbations F Launch Considerations

INJECTION REQUIREMENTS Speed

5 m 8 km

INJECTION REQUIREMENTS Speed 100 miles 17,500 mi/hr

INJECTION REQUIREMENTS Altitude Are you moving FASTER or SLOWER the higher your altitude?

INJECTION REQUIREMENTS Altitude V C = G(m 1 +m 2 ) aV C = 5.59 km/s V C = 4.56 km/s 2E 3E

INJECTION REQUIREMENTS Altitude 2E V E = V C 2 = 7.91 km/s V C = 5.59 km/s V < 7.91 km/s V > 7.91 km/s

INJECTION REQUIREMENTS Direction

ORBITAL MECHANICS Lesson 1 F Origins F Physical Laws F Requirements for Injection F Classifications of Orbits F Coordinate Reference Systems F Orbital Elements F Ground Tracks F Perturbations F Launch Considerations

ORBIT CLASSIFICATION F Size/Period F Location F Shape

ORBIT CLASSIFICATION Size/Period F Defined by semi-major axis (a) F Low Earth Orbit (LEO) F High Earth Orbit (HEO) F Semi-synchronous Orbit F Geo-synchronous Orbit

ORBIT CLASSIFICATION Location F Equatorial F Polar

ORBIT CLASSIFICATION Shape (Conic Sections) Circle Ellipse

ORBIT CLASSIFICATION Shape (Conic Sections) Trajectories: Parabola Hyperbola

ORBIT CLASSIFICATIONS Circular Orbits F Characteristics – Constant speed – Nearly constant altitude F Typical Missions – Reconnaissance/Weather (DMSP) – Manned – Navigational (GPS) – Geo-synchronous (Comm sats)

ORBIT CLASSIFICATIONS Elliptical Orbits F Characteristics – Varying speed – Varying altitude – Asymmetric Ground Track F Typical Missions – Deep space surveillance (Pioneer) – Communications (Polar comm.) – Ballistic Missiles

ORBIT CLASSIFICATIONS Parabolic/Hyperbolic Trajectories F Characteristics – Escaped Earth’s gravitational influence – Heliocentric F Typical Missions – Interplanetary exploration (Galileo, Phobos, Magellan)

ORBIT CLASSIFICATIONS Orbit Geometry Apogee Perigee cc a Eccentricity = c/a

ORBIT CLASSIFICATIONS Eccentricity e = 0 0 < e < 1 e = 1 e > 1

ORBIT CLASSIFICATIONS Eccentricity e = 0 a c = 0 0 < e < 1 c a Eccentricity = c/a

ORBIT CLASSIFICATIONS Eccentricity Eccentricity = c/a e = 0.75 e =.45 e = 0

ORBITAL MECHANICS Lesson 1 F Origins F Physical Laws F Requirements for Injection F Classifications of Orbits F Coordinate Reference Systems F Orbital Elements F Ground Tracks F Perturbations F Launch Considerations

COORDINATE SYSTEMS F Defines positions and directions in a consistent manner -- allows communication F Facilitates the description of a satellite’s position and subsequent motion F Proper choice of reference determines the utility of a coordinate system

COORDINATE SYSTEMS Ordinates F Origin – Where you’re starting from F Fundamental Plane – Plane which you’re measuring in F Principle Direction – Direction which you’re measuring from

COORDINATE SYSTEMS Classifications F Inertial – Non-rotating – Time Independent F Non-inertial – Rotating – Time Dependent

COORDINATE SYSTEMS Examples F Geographic F Topocentric F Geocentric Inertial F Orbit Inertial

COORDINATE SYSTEMS Geographic F Purpose: To locate points on the Earth’s surface

COORDINATE SYSTEMS Topocentric F Purpose: To locate a satellite with respect to a specific point on the Earth

COORDINATE SYSTEMS Topocentric Elevation

COORDINATE SYSTEMS Topocentric Azimuth Range Origin: Antenna FP: Local Horizon PD: True North

COORDINATE SYSTEMS Geocentric Inertial F Purpose: To determine the exact orientation of an orbital plane and to locate points in space with respect to the Earth Vernal Equinox Equatorial Plane Ecliptic Plane

COORDINATE SYSTEMS Geocentric Inertial Inclination

COORDINATE SYSTEMS Geocentric Inertial Vernal Equinox Ascending Node Direction of Satellite motion Right Ascension

COORDINATE SYSTEMS Orbit Inertial F Purpose: To fix the satellite orbit in the orbital plane Argument of Perigee Ascending Node Perigee

COORDINATE SYSTEMS Review F Geographic – Locates a point on the Earth’s surface – Requires Latitude and Longitude F Topocentric – Locates a satellite with respect to a site – Requires Azimuth, Elevation, Range

COORDINATE SYSTEMS Review F Geocentric Inertial – Locates orbital plane with respect to the Earth – Requires Right Ascension and Inclination F Orbit Inertial – Locate orbit within orbital plane – Requires Argument of Perigee

ORBITAL MECHANICS Lesson 1 F Origins F Physical Laws F Requirements for Injection F Classifications of Orbits F Coordinate Reference Systems F Orbital Elements F Ground Tracks F Perturbations F Launch Considerations

ORBITAL ELEMENTS Definition F A set of mathematical parameters that enables us to accurately describe satellite motion

ORBITAL ELEMENTS Purpose F Discriminate one satellite from other satellites F Predict where a satellite will be in the future or has been in the past F Determine amount and direction of maneuver or perturbation

ORBITAL ELEMENTS Keplerian Elements F Semi-Major Axis (Size) F Eccentricity (Shape) F Inclination F Right Ascension F Argument of Perigee F Epoch Time (Location within orbit) – True Anomaly (Orientation)

ORBITAL ELEMENTS Keplerian Elements: Inclination Orbital Plane Equatorial Plane Inclination ( i )

ORBITAL ELEMENTS Keplerian Elements: Right Ascension i Line of Nodes Right Ascension of the Ascending Node (  ) First Point of Aries (  )

ORBITAL ELEMENTS Keplerian Elements: Argument of Perigee  i Line of Nodes  Argument of Perigee (  )

ORBITAL ELEMENTS Keplerian Elements: True Anomaly True Anomaly ( ) Direction of satellite motion

ORBITAL ELEMENTS Keplerian Elements: True Anomaly i Line of Nodes   

ORBITAL ELEMENTS Keplerian Elements: Inclination Equatorial: i = 0 or 180 Polar: i = 90 Prograde: 0  i < 90 Retrograde: 90  i ú 180

ORBITAL MECHANICS Lesson 1 F Origins F Physical Laws F Requirements for Injection F Classifications of Orbits F Coordinate Reference Systems F Orbital Elements F Ground Tracks F Perturbations F Launch Considerations

Ground Tracks Westward Regression - Earth rotates east under a satellite => satellite appears to walk west - Earth rotates 360 degrees in 24 hours (15 degrees per hour)

Ground Tracks Westward Regression AB C A - time zero B - after one orbit C - after two orbits 60

Ground tracks Inclination N 45S Inclination = 45 degrees Eccentricity ~ 0

Ground Tracks Eccentricity Ground Track for Molnyia orbit eccentricity =.7252

ORBITAL MECHANICS Lesson 1 F Origins F Physical Laws F Requirements for Injection F Classifications of Orbits F Coordinate Reference Systems F Orbital Elements F Ground Tracks F Perturbations F Launch Considerations

PERTURBATIONS F Definition – A disturbance in the regular motion of a satellite F Types – Gravitational – Atmospheric Drag – Third Body Effects – Solar Wind/Radiation Effects – Electro-magnetic

PERTURBATIONS Gravitational: Libration F Ellipticity of the Earth causes gravity wells and hills F Stable points: 75E and 105W -- Himalayas and Rocky Mountains F Unstable points: 165E and 5W -- Marshall Islands and Portugal F Drives the requirement for stationkeeping

PERTURBATIONS Electro-Magnetic F Interaction between the Earth’s magnetic field and the satellite’s electro-magnetic field results in magnetic drag

ORBITAL MECHANICS Lesson 1 F Origins F Physical Laws F Requirements for Injection F Classifications of Orbits F Coordinate Reference Systems F Orbital Elements F Ground Tracks F Perturbations F Launch Considerations

LAUNCH CONSIDERATIONS Launch Windows F The period of time during which a satellite can be launched directly into a specific orbital plane from a specific launch site F Window duration driven by safety, fuel requirements, desired injection points, etc. F Window is centered around optimal launch time

PLACING SATELLITES IN ORBIT F Booster Types DELTA II

PLACING SATELLITES IN ORBIT F Booster Types ATLAS 2AS

PLACING SATELLITES IN ORBIT F Booster Types TITAN IV

PLACING SATELLITES IN ORBIT F Booster Types TAURUS

PLACING SATELLITES IN ORBIT F Booster Types The SHUTTLE BOOSTER

PLACING SATELLITES IN ORBIT F Booster Types PEGASUS

PLACING SATELLITES IN ORBIT F Launch Constraints