gn/mae1551 orbital mechanics overview 3 mae 155 g. nacouzi
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GN/MAE155 1
Orbital Mechanics Overview 3
MAE 155
G. Nacouzi
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Orbital Mechanics Overview 2
• Interplanetary Travel Overview– Coordinate system– Simplifications
• Patched Conic Approximation– Simplified Example– General Approach
• Gravity Assist: Brief Overview• Project Workshop
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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) = 149.6 E6 km (Pluto, a ~ 39.6 AU)
Heliocentric-ecliptic coordinate system for interplanetary transfer: Origin-center of Sun; fundamental plane- eclipticplane, principal direction - vernal equinox
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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
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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
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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
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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
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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
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Simple Example• Required velocities
calculated using fundamental orbital equations discussed earlier– ra= 149.6 km;
Va = 27.3 km/s
– rp= 108.2 km; Vp = 37.7 km/s
– Time of transfer ~146 days
Earth @Launch
Venus @arrival
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Simple Example• Example calculations:
Hyperbolic Excess Velocity, VHE ,S/C velocity wrt Earth
VHE = VS/S - VE/S
where,
VS/S ~ vel of S/C wrt Sun, VE/S ~ Vel Earth wrt Sun
VHE = 27.29 - 29.77 = -2.48 km/s
Similarly, @ target planet, hyperbolic excess vel, VHP needs to be
accounted for
VHP = VS/S - VP/S = 37.7 - 35 = 2.7 km/s; VP/S ~ Vplanet wrt Sun
Note: C3 = VHE2, Capability measure of LV
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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’
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Patched Conic Procedure• 1) Launch opportunity
To minimize required launch energy, Earth is placed (@ launch) directly opposed to tgt planet @ 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
Synodic period~ period between launch opport., S = 2pi/(e - target)S = 2yrs for Mars
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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
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Patched Conic Procedure• 3) Design Departure trajectory to escape Earth SOI, the departure
must be hyperbolic
where, Rpark~ parking orbit, VHE~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
Vearth
2
Rpark
he he
Vhe 2
2
V 2
2 V 2
R
park
Vretro
Vpark V
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Patched Conic Procedure
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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
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Gravity Assist Description
• The relative velocity of the SC can be increased or decreased depending on the approach trajectory
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Examples and Discussion