Interplanetary Trajectories

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Chapter 11

• Interplanetary Trajectories

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Patched conics(Hohmann1925)
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Patched conics

• Homann transfers dont take into account other
  gravity fields (2-Body approach)
• Interplanetary mission goes through different
  phases, each under a different gravity field.
• Instead of a single 2-body approach, use several
  2-body solutions along with the idea of a sphere
  of influence

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Universal Law of Gravitation
ag 0 as r
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Sphere of activity
r
R
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From the Sun-Earth two-body problem,
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Assume that r ltlt R
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The empirical formula usually used is
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Earths SOA(Sun-earth system)

• Earth m 5.981024 kg
• Sun M 1.991030 kg
• Astronaumical unit (AU)
• R 1.496 108 km
• r/R (m/3M)1/3 0.0100 , r 1,496,000 km
• r/R (m/M)2/5 0.0062 , r 924,850 km
• Only the order of magnitude matters.

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Planetary data
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Rendezvous
rendezvous point
U C F
Goal 20 yd pass Vreceiver 4 yd/s, Vball 10
yd/s Time of fly (TOF) 20 yd/(10 yd/s) 2
s Lead distance 4 yd/s 2 s 8 yd Head start
distance 20 yd - 8 yd 12 yd Wait time 12
yd/(4 yd/s) 3 s
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Launch window(Outbound)
Mars, arrival
q21
Earth, arrival
q12
Earth, departure
Mars, departure
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Transfer orbit
Time of fly
Phase angle at launch
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Launch window(Inbound)
Mars, departure
q21
Earth, departure
Earth, arrival
Mars, arrival
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Synodic period
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Synodic periods and trip times(days)
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Interplanetary trajectory
Earth
Earth parking orbit
Outside Earth SOA
Outside planet SOA
Planet parking orbit
Planet
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Assumptions
Radii of the planet orbit and the transfer orbit
Radii of the Earth and planet sphere activity
Radii of the Earth and planet parking orbits
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Results

• From the Earth parking orbit to the outside of
  the Earth SOA is considered as escape to
  infinity.
• After escape, the spacecraft is considered at
  the Earth orbit (around the Sun).
• The absolute velocity of the spacecraft after
  escape is the velocity of Earth plus the
  residual velocity after escape.

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Velocity needed
Hohmann transfer orbit from earth (planet 1) to
planet 2
At perihelion
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Velocities relative to earth
Residual velocity
Conservation of energy
Velocity after boost at parking orbit
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Dv
Velocity at parking orbit
The maneuver is
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Earth to Mars (Dv)
V 32.730 km/s, VÅ 29.785 km/s v 2.945
km/s Assume parking orbit altitude 200 km r0
6,578.137 km, v 7.784 km/s v0 11.40 km/s
(after boost) Dv 3.611 km/s
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Earth to Mars (time)
T12 2.2366107 s 258.8 days q12 44.3 q21
255.1 Twait 3.9256107 s 454.3 days Ttrip
2T12 Twait 972.1 days
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Escape from parking orbit
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Hyperbolic orbit
e gt 1, a lt 0, E gt 0
d
r ,
excess speed
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Keplers eq.hyperbola orbit
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Time to escape

• v 2.945 km/s, E v2/2 4.3365 km2/s2
• a -mÅ/(2E) -45,958.7 km
• r0 rp 6,578.137 km a(1 - e)
• e 1.143
• After escape, r 929,009.8 km
• 2.609 radians, F 3.6138 radians
• t T0 274,306 s 3.17 days

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Impact parameter
r , cosn - 1/e
y cos-1(-1/e) 151
b (-a r0)sin (180 - y) 25,470 km
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Possible maneuver points
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Timeline Earth to Mars

• Wait for the launch window (Mars 44.3 ahead of
  Earth)
• Escape from earth, 3.2 days
• Flying from Earth to Mars, 258.8 days
• Wait for coming back, 454.3 days
• Flying from Mars to Earth, 258.8 days
• Return to Earth, 3.2 days

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Planetary flyby
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Possible trajectories
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Optimal planetary capturing

• Given v, find the parking orbit so that the Dv
  is minimized.

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Capturing at Mars
The transfer orbit at 1.8877108 km At
aphelion, Voh 21.48 km/s v 24.13 - 21.48
2.65 km/s Dv 1.87 km/s Mars Gravitational
parameter mmars 4.269104 km3/s2 rp 24,316
km parking orbit altitude 24,316 3,395
20,921 km