Orbital Mechanics OverviewMAE 155ADr. George Nacouzi

James Webb Space Telescope, Launch Date 2011Primary mirror: 6.5-meter aperture Orbit: 930,000 miles from Earth , Mission lifetime: 5 years (10-year goal)Telescope Operating temperature: ~45 Kelvin Weight: Approximately 6600kg

Overview: Orbital MechanicsStudy of S/C (Spacecraft) motion influenced principally by gravity. Also considers perturbing forces, e.g., external pressures, on-board mass expulsions (e.g, thrust)Roots date back to 15th century (& earlier), e.g., Sir Isaac Newton, Copernicus, Galileo & KeplerIn early 1600s, Kepler presented his 3 laws of planetary motionIncludes elliptical orbits of planetsAlso developed Keplers eqtn which relates position & time of orbiting bodies

Overview: S/C Mission DesignInvolves the design of orbits/constellations for meeting Mission Objectives, e.g., area coverage Constellation design includes: number of S/C, number of orbital planes, inclination, phasing, as well as orbital parameters such as apogee, eccentricity and other key parameters Orbital mechanics provides the tools needed to develop the appropriate S/C constellations to meet the mission objectives

Introduction: Orbital MechanicsMotion of satellite is influenced by the gravity field of multiple bodies, however, Two body assumption is usually sufficient. Earth orbiting satellite Two Body approach: Central body is earth, assume it has only gravitational influence on S/C, assume M >> m (M, m ~ mass of earth & S/C)Gravity effects of secondary bodies including sun, moon and other planets in solar system are ignored Solution assumes bodies are spherically symmetric, point sources (Earth oblateness can be important and is accounted for in J2 term of gravity field)Only gravity and centrifugal forces are present

Two Body Motion (or Keplerian Motion)Closed form solution for 2 body exists, no explicit soltn exists for N >2, numerical approach neededGravitational field on body is given by:Fg = M m G/R2 where,

M~ Mass of central body; m~ Mass of Satellite

G~ Universal gravity constant R~ distance between centers of bodies

For a S/C in Low Earth Orbit (LEO), the gravity forces are:

Earth: 0.9 g Sun: 6E-4 g Moon: 3E-6 g Jupiter: 3E-8 g

General Two Body Motion EquationswhereSolution is in form of conical section, i.e., circle, ellipse, parabola & hyperbola.KE + PE, PE = 0 at R= a~ semi major axis of ellipseH = R x V = R V cos (), where H~ angular momentum & ~ flight path angle (between V & local horizontal) & r ~Position vectorV

General Two Body Motion TrajectoriesCentral BodyCircle, a=rEllipse, a> 0Hyperbola, a< 0Parabola, a = a Parabolic orbits provide minimum escape velocity Hyperbolic orbits used for interplanetary travel

General Solution to Orbital EquationVelocity is given by:

Eccentricity: e = c/a where, c = [Ra - Rp]/2 Ra~ Radius of Apoapsis, Rp~ Radius of Periapsis e is also obtained from the angular momentum H as: e = [1 - (H2/a)]; and H = R V cos ()

Circular Orbits EquationsCircular orbit solution offers insight into understanding of orbital mechanics and are easily derived Consider: Fg = M m G/R2 & Fc = m V2 /R (centrifugal F) V is solved for to get: V= (MG/R) = (/R) Period is then: T=2R/V => T = 2(R3/)

FcFgVR

Elliptical Orbit Geometry & NomenclaturePeriapsisApoapsisLine of ApsidesRacVRpb Line of Apsides connects Apoapsis, central body & Periapsis Apogee~ Apoapsis; Perigee~ Periapsis (earth nomenclature)S/C position defined by R & , is called true anomalyR = [Rp (1+e)]/[1+ e cos()]

Elliptical Orbit DefinitionOrbit is defined using the 6 classical orbital elements including:Eccentricity, semi-major axis, true anomaly and inclination, whereInclination, i, is the angle between orbit plane and equatorial planeiOther 2 parameters are: Argument of Periapsis (). Ascending Node: Pt where S/C crosses equatorial plane South to North Longitude of Ascending Node ()~Angle from Vernal Equinox (vector from center of earth to sun on first day of spring) and ascending nodeVernal EquinoxAscendingNodePeriapsis

Some Orbit Types...Extensive number of orbit types, some common ones:Low Earth Orbit (LEO), Ra < 2000 kmMid Earth Orbit (MEO), 2000< Ra < 30000 kmHighly Elliptical Orbit (HEO) Geosynchronous (GEO) Orbit (circular): Period = time it takes earth to rotate once wrt stars, R = 42164 kmPolar orbit => inclination = 90 degreeMolniya ~ Highly eccentric orbit with 12 hr period (developed by Soviet Union to optimize coverage of Northern hemisphere)

Sample OrbitsLEO at 0 & 45 degree inclinationElliptical, e~0.46, I~65deg Ground tracefrom i= 45 deg

Sample GEO OrbitFigure 8 trace due to inclination, zero inclination in nomotion of nadir point(or satellite sub station) Nadir for GEO (equatorial, i=0) remain fixed over point 3 GEO satellites provide almost complete global coverage

Orbital Maneuvers DiscussionOrbital ManeuverS/C uses thrust to change orbital parameters, i.e., radius, e, inclination or longitude of ascending nodeIn-Plane Orbit ChangeAdjust velocity to convert a conic orbit into a different conic orbit. Orbit radius or eccentricity can be changed by adjusting velocityHohmann transfer: Efficient approach to transfer between 2 Non-intersecting orbits. Consider a transfer between 2 circular orbits. Let Ri~ radius of initial orbit, Rf ~ radius of final orbit. Design transfer ellipse such that: Rp (periapsis of transfer orbit) = Ri (Initial R) Ra (apoapsis of transfer orbit) = Rf (Final R)

Hohmann Transfer DescriptionDV1DV2TransferEllipseFinal OrbitInitial OrbitRp = RiRa = RfDV1 = Vp - ViDV2 = Va - Vf

Note:( )p = transfer periapsis( )a = transfer apoapsisRpRaRiRf

General In-Plane Orbital Transfers...Change initial orbit velocity Vi to an intersecting coplanar orbit with velocity Vf DV2 = Vi2 + Vf2 - 2 Vi Vf cos (a)Initial orbitFinal orbitaVfDVVi

Other Orbital Transfers...Bielliptical TranferWhen the transfer is from an initial orbit to a final orbit that has a much larger radius, a bielliptical transfer may be more efficientInvolves three impulses (vs. 2 in Hohmann)Plane ChangesCan involve a change in inclination, longitude of ascending nodes or bothPlane changes are very expensive (energy wise) and are therefore avoided if possible

Basics of Rocket EquationF = Ve dm/dtF = M dV/dtThrustdV/dtVe ~ Exhaust Vel.m ~ propellant massF = Thrust = ForceM ~S/C MassV ~ S/C Velocitygc~ gravitational constantS/CM dV/dt = Ve dm/dt = - Ve dM/dt

=> DV = Ve ln (Mi/Mf) where, Mi ~ Initial Mass; Mf~ Final Mass

Isp = Thrust/(gc dm/dt) => Ve = Isp x gcCalculate mass of propellant needed for rocket to provide a velocitygain (DV)

Basics of Rocket Equation (contd)M dV/dt = Ve dm/dt = - Ve dM/dt

=> DV = Ve ln (Mi/Mf) where, Mi ~ Initial Mass; Mf~ Final Mass

Isp = Thrust/(gc dm/dt) => Ve = Isp x gcSubstituting we get:Mi/Mf = exp (DV/ (gc Isp))but Mp = Mf - Mi =>Mp = Mi[1-exp(-DV/ gc Isp)]Where, DV ~ Delta Velocity, Mp ~ Mass of PropellantMass of propellant calculated from Delta Velocity and propellant Isp.For Launch Vehicles: Isp ~ 260 - 300 sec for solid propellant Isp ~ 300 - 500 sec for liquid bipropellant

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