Orbital Mechanics Overview MAE 155A G. Nacouzi GN/MAE155A

James Webb Space Telescope, Launch Date 2011 Primary mirror: 6.5-meter aperture Orbit: 930,000 miles from Earth , L3 point Mission lifetime: 5 years (10-year goal) Telescope Operating temperature: ~45 Kelvin Weight: Approximately 6600kg GN/MAE155A

Overview: Orbital Mechanics Study 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 & Kepler In early 1600s, Kepler presented his 3 laws of planetary motion Includes elliptical orbits of planets Also developed Kepler’s eqtn which relates position & time of orbiting bodies Kepler's three laws of planetary motion can be described as follows: The path of the planets about the sun are elliptical in shape, with the center of the sun being located at one focus. (The Law of Ellipses) An imaginary line drawn from the center of the sun to the center of the planet will sweep out equal areas in equal intervals of time. (The Law of Equal Areas) The ratio of the squares of the periods of any two planets is equal to the ratio of the cubes of their average distances from the sun. (The Law of Harmonies) GN/MAE155A

Overview: S/C Mission Design Involves the design of orbits/constellations for meeting Mission Objectives, e.g., coverage area 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 GN/MAE155A

Introduction: Orbital Mechanics Motion of satellite is influenced by the gravity field of multiple bodies, however, 2 body assumption is usually used for initial studies. Earth orbiting satellite 2 Body assumptions: Central body is Earth, assume it has only gravitational influence on S/C, MEarth >> mSC 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 GN/MAE155A

Sources of Orbital Perturbations Several external forces cause perturbation to spacecraft orbit 3rd body effects, e.g., sun, moon, other planets Unsymmetrical central bodies (‘oblateness’ caused by rotation rate of body): Earth: Radius at equator = 6378 km, Radius at polar = 6357 km Space Environment: Solar Pressure, drag from rarefied atmosphere GN/MAE155A

Relative Importance of Orbit Perturbations Reference: Spacecraft Systems Engineering, Fortescue & Stark J2 term accounts for effect from oblate earth Principal effect above 100 km altitude Other terms may also be important depending on application, mission, etc... GN/MAE155A

Two Body Motion (or Keplerian Motion) Closed form solution for 2 body exists, no explicit solution exists for N >2, numerical approach needed Gravitational 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 GN/MAE155A

Two Body Motion (Derivation) h r M m j For m, we have m.h’’ = GMmr/(r^2 |r|) m.h’’ = GMmr/r^3 h’’ = r/r^3 where h’’= d2h/dt2 &  = GM For M, Mj’’ = -GMmr/(r^2 |r|) j’’ = -Gmr/r^3, but r = j-h => r’’ = -G(M+m) r/r^3 for M>>m => r’’ + GM r/r^3= 0, or r’’ + r/r^3 = 0 (1) GN/MAE155A

Two Body Motion (Derivation) From r’’ + r/r^3 = 0 => r x r’’ + r x r/r^3 = 0 => r x r’’ = 0, but r x r’’ = d/dt ( r x r’) = d/dt (H),  d/dt (H) =0, where H is angular momentum vector, i.e. r and r’ are in same plane. Taking the cross product of equation 1with H, we get: (r’’ x H) + /r^3 (r x H) = 0 (r’’ x H) = /r^3 (H x r), but d/dt (r’ x H) = (r’’x H) + (r’ x H’) => d/dt (r’ x H) = /r^3 (H x r) => d/dt (r’ x H) = /r^3 (r2 ’) r  =  ’  =  r’ ( r is unit vector)  d/dt (r’ x H) =  r’ ; integrate => r’ x H =  r + B =0 H= r2 ’  is a unit vector normal to the unit vector r along the r vector GN/MAE155A

Two Body Motion (Derivation) r . (r’ x H) = r . ( r + B) = (r x r’) . H = H.H = H2 => H2 = r + r B cos () => r = (H2 / )/[1 + B/ cos()] p = H2 / ; e = B /  ~ eccentricity;  ~ True Anomally => r = p/[1+e cos()] ~ Equation for a conic section where, p ~ semilatus rectum Specific Mechanical Energy Equation is obtained by taking the dot product of the 2 body ODE (with r’), and then integrating the result r’.r’’ + r.r’/r^3 = 0, integrate to get: r’2/2 - /r =  B is the constant of integration P (semilatus rectum) is the the normal distance from the focus point to the ellipse p GN/MAE155A

General Two Body Motion Equations d2r/dt2 +  r/R3 = 0 (1) where,  = GM, r ~Position vector, and R = |r| Solution is in form of conical section, i.e., circle ~ e = 0, ellipse ~ e < 1 (parabola ~ e = 1 & hyperbola ~ e >1) V Specific mechanical energy is:  Local Horizon KE + PE, PE = 0 at R=  & PE<0 for R<  Potential energy is considered to be 0 at infinity and negative at distances less than infinity a~ semi major axis of ellipse H = R x V = R V cos (), where H~ angular momentum &  ~ flight path angle (FPA, between V & local horizontal) GN/MAE155A

Circular Orbits Equations Circular 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=2R/V => T = 2(R3/) V Fc R Fg * Period = time it takes SC to rotate once wrt earth GN/MAE155A

General Two Body Motion Trajectories Hyperbola, a< 0 a Circle, a=r Parabola, a =  Ellipse, a > 0 Central Body Parabolic orbits provide minimum escape velocity Hyperbolic orbits used for interplanetary travel GN/MAE155A

Elliptical Orbit Geometry & Nomenclature V Periapsis a c R  Line of Apsides Rp Apoapsis b S/C position defined by R & ,  is called true anomaly R = [Rp (1+e)]/[1+ e cos()] Line of Apsides connects Apoapsis, central body & Periapsis Apogee~ Apoapsis; Perigee~ Periapsis (Earth nomenclature) GN/MAE155A

Elliptical Orbit Definition Orbit is defined using the 6 classical orbital elements including: Eccentricity, semi-major axis, true anomaly and inclination, where Inclination, i, is the angle between orbit plane and equatorial plane Periapsis i  Vernal Equinox  Ascending Node Other 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 node GN/MAE155A

General Solution to Orbital Equation Velocity 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 () GN/MAE155A

More Solutions to Orbital Equation FPA is given by: tan() = e sin()/ ( 1+ e cos()) True anomaly is given by, cos() = (Rp * (1+e)/R*e) - 1/e Time since periapsis is calculated as: t = (E- e sin(E))/n, where, n = /a3; E = acos[ (e+cos())/ ( 1+ e cos()] E is called the eccentric anomaly, n is the mean motion these are circular equivalent parameters to elliptical motion GN/MAE155A

Some Orbit Types... Extensive number of orbit types, some common ones: Low Earth Orbit (LEO), Ra < 2000 km Mid Earth Orbit (MEO), 2000< Ra < 30000 km Highly Elliptical Orbit (HEO) Geosynchronous (GEO) Orbit (circular): Period = time it takes earth to rotate once wrt stars, R = 42164 km Polar orbit => inclination = 90 degree Molniya ~ Highly eccentric orbit with 12 hr period (developed by Soviet Union to optimize coverage of Northern hemisphere) GN/MAE155A

Sample Orbits LEO at 0 & 45 degree inclination Elliptical, e~0.46, I~65deg Lat =.. Ground trace from i= 45 deg GN/MAE155A

Sample GEO Orbit Nadir for GEO (equatorial, i=0) remain fixed over point 3 GEO satellites provide almost complete global coverage Figure ‘8’ trace due to inclination, zero inclination has no motion of nadir point (or satellite sub station) GN/MAE155A

Orbital Maneuvers Discussion S/C uses thrust to change orbital parameters, i.e., radius, e, inclination or longitude of ascending node In-Plane Orbit Change Adjust velocity to convert a conic orbit into a different conic orbit. Orbit radius or eccentricity can be changed by adjusting velocity Hohmann 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) GN/MAE155A

Hohmann Transfer Description Ellipse Rp = Ri Ra = Rf DV1 = Vp - Vi DV2 = Va - Vf DV = |DV1|+|DV2| Note: ( )p = transfer periapsis ( )a = transfer apoapsis DV1 Ra Rp Ri Rf Initial Orbit DV2 Final Orbit GN/MAE155A

General In-Plane Orbital Transfers... Change initial orbit velocity Vi to an intersecting coplanar orbit with velocity Vf (basic trigonometry) DV2 = Vi2 + Vf2 - 2 Vi Vf cos (a) Final orbit DV Initial orbit Vi Vf a GN/MAE155A

Aerobraking Aerobraking uses aerodynamic forces to change the velocity of the SC therefore its trajectory (especially useful in interplanetary missions) Instead of retro burns, aero forces are used to change the vehicle velocity GN/MAE155A

Other Orbital Transfers... Hohman transfers are not always the most efficient Bielliptical Tranfer When the transfer is from an initial orbit to a final orbit that has a much larger radius, a bielliptical transfer may be more efficient Involves three impulses (vs. 2 in Hohmann) Low Thrust Transfers When thrust level is small compared to gravitational forces, the orbit transfer is a very slow outward spiral Gravity assists - Used in interplanetary missions Plane Changes Can involve a change in inclination, longitude of ascending nodes or both Plane changes are very expensive (energy wise) and are therefore avoided if possible GN/MAE155A

Examples & Announcements GN/MAE155A