Basic Orbital Mechanics Dr. Andrew Ketsdever MAE 5595

Conic Sections EccentricityConic = 0Circle 0 - 1Ellipse = 1Parabola > 1Hyperbola

Elliptical Orbit Geometry

Conic Sections

Classical Orbital Elements Semi-Major Axis, a –Size Eccentricity, e –Shape Kepler’s 3 rd Law

Classical Orbital Elements Inclination –Tilt InclinationOrbit = 90ºPolar 0º or 180ºEquatorial 0º - 90ºPrograde 90º - 180ºRetrograde

Classical Orbital Elements Right Ascension of the Ascending Node (RAAN)

Classical Orbital Elements Argument of Perigee

Classical Orbital Elements True Anomaly

Computing COEs From a R and V vector –Can compute the 6 COEs –Also works in reverse (given COEs compute R and V) –Example:

COEs a = km e = i = 90º  = 270º  = 90º = 0º Mission: Probably remote sensing or a spy satellite because it’s in a low, polar orbit.

Ground Tracks Ground Track Slides Courtesy of Major David French

COE Determination Semimajor axis Δ longitude ΔNΔN P = ΔN 15º / hr

COE Determination Eccentricity

COE Determination Inclination i = highest latitude

COE Determination Argument of Perigee ω = 90º

COE Determination True Anomaly

Orbit Examples

Molniya

Geostationary

Geosynchronous

e = 0 i = 0  Geosynchronous e = 0.4  = 180  e = 0.6  = 90 

Orbit Prediction Kepler’s Problem –If we know where a satellite (or planet) is today, where in its orbit will it be tomorrow? –Kepler devised a series of mathematical expressions to solve this particular problem Eccentric Anomaly Mean Anomaly True Anomaly II. The line joining the planet to the Sun sweeps out equal areas in equal times as the planet travels around the ellipse.

Orbit Prediction Kepler defined the Eccentric Anomaly to relate elliptical motion to circular motion He also defined Mean Anomaly to make the circular motion constant Convert unsteady elliptical motion into unsteady circular motion into steady circular motion…

Orbit Prediction

Orbital Prediction Given a = 7000 km e = 0.05 = 270º Find the time of flight to final = 50º

Orbital Prediction n = rad/sec E initial = º E future = 47.84º M initial = º M future = 45.72º TOF = sec or min