GN/MAE1551 Orbital Mechanics Overview 3 MAE 155 G. Nacouzi

GN/MAE1552 Orbital Mechanics Overview 2 Interplanetary Travel Overview –Coordinate system –Simplifications Patched Conic Approximation –Simplified Example –General Approach Gravity Assist: Brief Overview Project Workshop

GN/MAE1553 Interplanetary Travel: Coordinate System Use Heliocentric coordinate system,i.e., Sun centered –Use plane of earth’s orbit around sun as fundamental plane, called ecliptic plane –Principal direction, I, is in vernal equinox Define 1 Astronomical Unit (AU) = semimajor axis of earth orbit (a) = E6 km (Pluto, a ~ 39.6 AU) Heliocentric-ecliptic coordinate system for interplanetary transfer: Origin- center of Sun; fundamental plane- ecliptic plane, principal direction - vernal equinox

GN/MAE1554 Principal Forces in Interplanetary Travel In addition to forces previously discussed, we now need to account for Sun & target planet gravity Simplify to account only for gravity forces, ignore solar pressure & other perturbances in mission planning => forces considered - Gravity effects of Sun, Earth, Target Planet on S/C

GN/MAE1555 Principal Forces in Interplanetary Travel A very useful approach in solving the interplanetary trajectory problem is to consider the influence on the S/C of one central body at a time => Familiar 2 body problem –Consider departure from earth: 1) Earth influence on vehicle during departure 2) Sun influence=> Sun centered transfer orbit 3) Target planet gravity influence, arrival orbit

GN/MAE1556 Interplanetary Trajectory Model: Patched Conic Divide interplanetary trajectory into 3 different regions => basis of the patched- conic approximation –Region 1: Sun centered transfer (Sun gravity dominates) - solved first –Region 2: Earth departure (Earth gravity dominates) - solved second (direct or from parking orbit) –Region 3: Arrival at target planet (planet gravity dominates) - solved third

GN/MAE1557 Interplanetary Trajectory Model: Patched Conic Two body assumption requires calculating the gravitational sphere of influence, Rsoi, of each planet involved Rsoi (radius) = a(planet) x (Mplanet/Msun)^0.4 Earth Rsoi = 1E6 km Rsoi gravity Sphere of Influence

GN/MAE1558 Interplanetary Trajectory Model: Patched Conic Sun centered transfer solved first since solution provides information to solve other 2 regions Consider simplified example of Earth to Venus –Assume circular, coplanar orbits (constant velocity and no plane change needed) –Hohmann transfer used: Apoapsis of transfer ellipse = radius of Earth Orbit Periapsis of transfer ellipse = radius of Venus Orbit

GN/MAE1559 Simple Example Required velocities calculated using fundamental orbital equations discussed earlier –r a = km; V a = 27.3 km/s –r p = km; V p = 37.7 km/s –Time of transfer ~146 days Launch arrival

GN/MAE15510 Simple Example Example calculations: Hyperbolic Excess Velocity, V HE,S/C velocity wrt Earth V HE = V S/S - V E/S where, V S/S ~ vel of S/C wrt Sun, V E/S ~ Vel Earth wrt Sun V HE = = km/s target planet, hyperbolic excess vel, V HP needs to be accounted for V HP = V S/S - V P/S = = 2.7 km/s; V P/S ~ Vplanet wrt Sun Note: C3 = V HE 2, Capability measure of LV

GN/MAE15511 Patched Conic Procedure 1) Select a launch date based on launch opportunity analysis 2) Design transfer ellipse from earth to tgt planet 3) Design departure trajectory (hyperbolic) 4) Design approach trajectory (hyperbolic) Reference: C. Brown, ‘Elements of SC Design’

GN/MAE15512 Patched Conic Procedure 1) Launch opportunity To minimize required launch energy, Earth is placed launch) directly opposed to tgt arrival –Calc. TOF, ~ 1/2 period of transfer orbit –Calculate lead angle = Earth angular Vel (  e )x TOF –Phase angle,  r = 2 pi - lead angle Wait time =  r -  current /(  target -  e ) Earth Mars rr Synodic period~ period between launch opport., S = 2pi/(  e -  target ) S = 2yrs for Mars

GN/MAE15513 Patched Conic Procedure 2) Develop transfer ellipse from Earth to Target Planet (heliocentric) accounting for plane change as necessary –Note that the transfer ellipse is on a plane that intersects the Sun & Earth at launch, & the target planet at arrival. Plane change usually made at departure to combine with injection and use LV energy instead of S/C

GN/MAE15514 Patched Conic Procedure 3) Design Departure trajectory to escape Earth SOI, the departure must be hyperbolic where, Rpark~ parking orbit, V HE ~hyperbolic excess velocity 4) Design approach trajectory to target planet where Vpark is the orbital velocity in the parking orbit and V  is the SC velocity at arrival. Vretro is the delta V to get into orbit

GN/MAE15515 Patched Conic Procedure

GN/MAE15516 Gravity Assist Description Use of planet gravity field to rotate S/C velocity vector and change the magnitude of the velocity wrt Sun. No SC energy is expended Reference: Elements of SC Design, Brown

GN/MAE15517 Gravity Assist Description The relative velocity of the SC can be increased or decreased depending on the approach trajectory

GN/MAE15518 Examples and Discussion