Basic Orbital Mechanics Jeff Crum 23 Aug 01 9/19/2018

Introduction Orbital mechanics is the study of the motions of artificial satellites and space vehicles moving under the influence of forces such as gravity, atmospheric drag, thrust, etc. Modern offshoot of celestial mechanics, the study of the motions of the moon, planets, and stars 9/19/2018

Training Topics Kepler and Newton Orbital Elements Ground Traces Orbit Types Orbit Perturbations 9/19/2018

Planetary motion The problem of accurately describing the motions of planets has challenged observers for centuries Early theories held that the Earth was the center of the universe and all other heavenly bodies traveled in perfect circles about the Earth In 1543, Nicolaus Copernicus published his heliocentric theory that postulated that the Sun was the center of the universe, and that all planets—including the Earth, revolved about it in perfect circles 9/19/2018

Kepler Johannes Kepler agreed with Copernicus’ revolutionary and highly controversial theory Using thousands of astronomical observations from the Danish astronomer Tycho Brahe, Kepler tried in vain to fit the motion of the planet Mars to a circular orbit Kepler finally discovered the answer: planets travel not in circles, but ellipses about the sun 9/19/2018

Ellipse GEOMETRIC CENTER Semi-Minor Axis (b) FOCUS FOCUS Semi-Major Axis (a) Focal Length (c) MAJOR AXIS MINOR AXIS Based on the results of his observations, Kepler published his three laws of planetary motion 9/19/2018

Kepler’s First Law The orbit of each planet is an ellipse, with the Sun at one focus 9/19/2018

Kepler’s Second Law The line joining the planet to the Sun sweeps out equal areas in equal times 9/19/2018

Kepler’s Third Law The square of a planet’s orbital period is in direct proportion to the cube of the semi-major axis Orbital period (P) = time required to make one complete revolution around the sun P2  a3 For example, Mercury, the closest planet to the Sun, completes an orbit in 88 days Pluto, the furthest planet from the Sun, completes an orbit every 248 years 9/19/2018

Newton Building upon the work of Kepler and others, Isaac Newton put forward his laws of motion and formulated his law of universal gravitation 9/19/2018

Newton’s Laws of Motion First Law: A body at rest remains at rest, and a body in motion continues to move at a constant velocity unless acted upon by an external force Second Law: A force (F) acting on a body gives it an acceleration (a) which is in the direction of the force and has a magnitude inversely proportional to the mass of the body (m) F = ma Third Law: Whenever a body exerts a force on another body, the latter exerts a force of equal magnitude and opposite direction on the former 9/19/2018

Newton in simpler terms Objects in motion want to travel in straight lines at constant speed (Newton’s first law) But… Force of gravity causes the path to curve 9/19/2018

Newton in simpler terms Amount of curve depends on initial speed and direction 9/19/2018

Newton in simpler terms Satellites must have a balance of… Speed + Gravity = No orbit Orbit No Orbit (too slow) (too fast) 9/19/2018

Satellite orbits The initial speed and direction of an orbiting satellite creates an ellipse with the Earth at one focus 9/19/2018

Satellite orbits Apogee = highest altitude, lowest speed Perigee = lowest altitude, fastest speed Perigee Apogee Note: In generic terms, periapsis is the point in an orbit closest to the primary focus, and apoapsis is the farthest point. These terms are usually modified to apply to the body being orbited (e.g., perihelion/aphelion for the Sun, perigee/apogee for the Earth, perijove/apojove for Jupiter, etc.) 9/19/2018

Satellite Orbits Orbit size determines satellite period, or time to make one orbit DMSP 510 miles 110 minutes GPS 10900 miles 11 hrs, 58 min DSP 22300 miles 23 hrs, 56 min 9/19/2018

Satellite Orbits Orbital element set Semi-Major Axis……………………….. Size Eccentricity………………………………. Shape Inclination……………………………….. Tilt Right Ascension of the Ascending Node……………………….. Direction Argument of Perigee…………………. Rotation True Anomaly…………………………… Position Epoch Time………………………….…… Time Stamp 9/19/2018

Semi-Major Axis (Size) Semi-Major Axis (a) FOCUS PERIGEE APOGEE Focal Length (c) Focal Length (c) 9/19/2018

Eccentricity (Shape) e = .75 e = .45 e = 0 Eccentricity (e) = c/a a  9/19/2018

Inclined Prograde Orbit Inclination (Tilt) Inclination Angle 0 to < 90  Equatorial Plane Ascending Node Inclined Prograde Orbit 9/19/2018

Inclined Retrograde Orbit Inclination (Tilt) Inclination Angle >90 to < 180  Equatorial Plane Ascending Node Inclined Retrograde Orbit 9/19/2018

Right Ascension of Ascending Node (Direction) Inclined Prograde Orbit 0 = First Point of Aries 0 180 Ascending Node Line of Nodes 9/19/2018

Right Ascension of Ascending Node (Direction) 5 GPS constellation has 6 orbit planes, all inclined at 55° 1 6 Each orbit plane is spaced 60° from the previous plane 2 4 3 9/19/2018

Argument of Perigee (Rotation) Argument of Perigee is the angular distance between the ascending node and the point of perigee Perigee Equatorial Plane Ascending Node 9/19/2018

True Anomaly True Anomaly is the angular distance between perigee and some point in the orbit. Usually this is used to describe where an SV is in the orbit at a given time. Equatorial Plane Perigee 9/19/2018

Epoch Time The epoch time is an exact specification of the date and time at which a given Keplerian element set is valid 9/19/2018

Ground Trace A ground trace is the projection of a satellite’s orbit onto the earth’s surface Note that the highest northern and southern latitudes reached by a satellite’s ground trace is equal to the satellite’s orbital inclination If inclination > 90°, then highest latitude of the ground trace is 180° minus the orbit inclination 9/19/2018

Ground Trace Consider an orbit of a satellite to lie in a plane that passes through the center of a theoretically spherical Earth The trace of this plane on the surface of a non-rotating Earth is a great circle If the Earth did not rotate, the satellite would retrace the same ground over and over 9/19/2018

Ground Trace If we consider a rotating Earth, the orbital plane of a satellite remains fixed in space as the Earth turns under it Effect of Earth’s rotation is to displace the ground trace on each successive revolution of the satellite Ground trace displaced by the number of degrees the Earth rotates during on orbital period This displacement is called nodal regression 9/19/2018

Nodal Regression Earth’s rotation causes the ground trace to regress westward for each successive orbit Earth rotates approximately 15° per hour Nodal regression = (orbit period in hours) * 15° A satellite in a polar orbit has the potential to overfly all the Earth’s surface If the time required for one complete rotation of the Earth is an exact multiple of the satellite’s period, then eventually the satellite will retrace exactly the same path as it did on some previous revolution 9/19/2018

GROUND TRACES Westward Regression Pictures and animation courtesy of Capt Troy Endicott, Det 1 533 TRS, Schriever AFB, AETC

Orbit Types Satellites use a wide variety of orbits to fulfill their missions Factors determining orbit type: Mission requirements Booster capability and cost Satellite and orbital mechanics 9/19/2018

Orbit Types For a spacecraft to achieve orbit, it must be launched to an elevation above the Earth’s atmosphere and accelerated to orbital velocity The most energy efficient orbit is a direct, low inclination orbit To achieve this orbit, the spacecraft is launched in an eastward direction from a site near the Earth’s equator The rotational speed of the Earth contributes to the spacecraft’s final orbital speed e.g. A due east launch from the Cape (28.5 deg north latitude) results in a “free ride” of 915 mph 9/19/2018

Orbit Types Launching a spacecraft in a direction other than east, or from a site far from the equator, results in an orbit of higher inclination High inclination orbits are less able to take advantage of the initial speed provided by the Earth’s rotation Launch vehicle must provide greater energy to attain orbital velocity 9/19/2018

Safety constrains possible launch azimuths The desired orbit inclination determines the azimuth of launch. Vandenberg AFB Patrick AFB Safety constrains possible launch azimuths 9/19/2018

Orbit Types Low Earth Orbit (LEO) Medium Earth Orbit (MEO) Sun Synchronous Polar Medium Earth Orbit (MEO) Geosynchronous/Geostationary Earth Orbit (GEO) Highly inclined orbits Molniya 9/19/2018

Low Earth Orbits Up to 520 miles Common missions: Manned (shuttle) Reconnaissance Communications 9/19/2018

LEO 9/19/2018

Polar Orbits Inclination of 90 degrees Missions: Mapping Surveillance 9/19/2018

Sun Synchronous 460-520 miles Near-polar inclination Orbital plane precesses with the same period as the Earth’s solar period Satellite crosses perigee at the same local time every orbit Common missions: Earth sensing (LANDSAT) Weather (DMSP/NOAA) 9/19/2018

Sun Synchronous 9/19/2018

Medium Earth Orbit Also called semi-synchronous 10,900 miles; high inclination Missions: Navigation (GPS, GLONASS) 9/19/2018

MEO 9/19/2018

Geosynchronous and Geostationary Altitude = 22,300 miles Period = 24 hours Any inclination Does not have to be circular Geostationary Inclination near zero Eccentricity nearly zero 9/19/2018

GEO GEO spacecraft appear to hang motionless above one position on the Earth’s surface Missions: Communications Weather 9/19/2018

GEO 9/19/2018

Highly Elliptical Orbits 63.4-116.6 degree inclination 200-23,800 mile altitude Missions: Comm relay (Molniya) 9/19/2018

Molniya Orbits Many Russian cities are at high northern latitudes where it is impractical to use GEO satellites for telecommunications GEO satellites appear either low on the horizon or are not visible at all Molniya orbit has a 12 hour period at high eccentricity and inclination (63.4°) Satellite spends most of its time near apogee, so for approximately 11 hours of each orbit the satellite is above the horizon for high northern latitudes 9/19/2018

Molniya Orbit 9/19/2018

GROUND TRACES Highly Elliptical Orbit Pictures and animation courtesy of Capt Troy Endicott, Det 1 533 TRS, Schriever AFB, AETC

Orbit Perturbations Any disturbance in the regular motion of a satellite resulting from a force other than those causing regular motion Non-spherical earth Atmospheric drag Sun/moon gravity Space environment 9/19/2018

Summary TBD 9/19/2018

Backup Charts 9/19/2018

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