Transcript Orbital Mechanics II: Transfers, Rendezvous, Patched Conics, and Perturbations Dr. Andrew Ketsdever

Orbital Mechanics II:
Transfers, Rendezvous, Patched
Conics, and Perturbations
Dr. Andrew Ketsdever
Lesson 3
MAE 5595
Orbital Transfers
• Hohmann Transfer
– Efficient means of increasing/decreasing orbit
size
– Doesn’t truly exist
– Assumptions
•
•
•
•
Initial and final orbits in the same plane
Co-apsidal orbits (Major axes are aligned)
ΔV is instantaneous
ΔV is tangential to initial and final orbits (velocity
changes magnitude but not direction)
Hohmann Transfer
Hohmann Transfer
Conceptual Walkthrough
alt1 = 300 km
alt2 = 1000 km
2
V1
ΔV1
Slides Courtesy of Major David French, USAFA/DFAS
2
Vt1
ΔV2
Vt2
2
V2
2
Time of Flight
3
Ptrans
atrans
TOF 

2

2
Hohmann Transfer
Orbital Transfers
• One Tangent Burn Transfer
– First burn is tangent to the initial orbit
– Second burn is at the final orbit
• Transfer orbit intersects final orbit
• An infinite number of transfer orbits exist
• Transfer orbit may be elliptical, parabolic or
hyperbolic
– Depends on transfer orbit energy
– Depends on transfer time scale
One-Tangent Burn
One-Tangent Burn
Spiral Transfer
Expect to multiply by as much as a
factor of 2 for some missions
Orbital Transfer
• Plane Changes
– Simple
• Only changes the inclination of the orbit, not its
size
– Combined
• Combines the ΔV maneuver of a Hohmann
(tangential) transfer with the ΔV maneuver for a
plane change
• Efficient means to change orbit size and inclination
Plane Changes
• Simple
–
• Combined
–
Rendezvous
• Co-Orbital Rendezvous
– Interceptor and Target initially in the same
orbit with different true anomalies
• Co-Planar Rendezvous
– Interceptor and Target initially in different
orbits with the same orbital plane (inclination
and RAAN)
Co-Orbital Rendezvous
Co-Orbital Rendezvous
Target Leading
Co-Orbital Rendezvous
Target Leading
Co-Orbital Rendezvous
Target Leading
3 step process for determining phasing orbit size
Co-Orbital Rendezvous
Target Leading
ωTGT
1
Co-Orbital Rendezvous
Target Leading
ωTGT
travel
TOF 
Tgt
Φtravel
2
Co-Orbital Rendezvous
Target Leading
ωTGT
Φtravel
TOF  2
a 3phase

3
Co-Orbital Rendezvous
Target Trailing
Co-Orbital Rendezvous
Target Trailing
Co-Orbital Rendezvous
Target Trailing
ωTG
T
Φtravel
Co-Planar Rendezvous
Coplanar Rendezvous
2
5 step process for determining wait time (WT)
1
ωTGT
ωINT
2
2
TOF
2
TOF  
3
transfer
a

3
ωTGT
TOF
αlead
ωINT
2
 lead  t arg etTOF
4
ωTGT
Φfinal
αlead
ωINT
2
 final     lead
5
ωTGT
Φfinal
αlead
ωINT
2
Φinitial
 final  initial
WT 
t arg et  int ercept
Interplanetary Travel
• In our two-body universe (based on the restricted,
two-body EOM), we can not account for the
influence of other external forces
– In reality we can account for many body problems, but
for our purposes of simplicity we will stick to two-body
motion in the presence of gravity
– Need a method to insure that only two-bodies are
acting during a particular phase of the spacecraft’s
motion
• Spacecraft – Earth (from launch out to the Earth’s SOI)
• Spacecraft – Sun (From Earth SOI through to the Target SOI)
• Spacecraft – Planet (From Target Planet SOI to orbit or
surface)
Patched Conic Approximation
• Spacecraft – Earth
– Circular or Elliptical low-Earth orbit (Parking)
– Hyperbolic escape
– Geo-centric, equatorial coordinate system
• Spacecraft – Sun
– Elliptical Transfer Orbit
– Helio-centric, ecliptic coordinate system
• Spacecraft – Target
– Hyperbolic arrival
– Circular or Elliptical orbit
– Target-centric, equatorial coordinate system
Patched Conic Approximation
Geo: Hyperbolic escape
Helio: Elliptical transfer
Targeto: Hyperbolic arrival
Orbital Perturbations
• Several factors cause perturbations to a
spacecraft’s attitude and/or orbit
– Drag
– Earth’s oblateness
– Actuators
– 3rd bodies
– Gravity gradient
– Magnetic fields
– Solar pressure
Orbital Drag
• Orbital drag is an issue in low-Earth orbit
– Removes energy from the s/c orbit (lowers)
– Orbital decay due to drag depends on several
factors
• Spacecraft design
• Orbital velocity
• Atmospheric density
– Altitude, Latitude
– Solar activity
FDrag
1
 C D V 2 AFrontal
2
3rd Bodies
• Geosynchronous Equatorial Orbits are
influenced by the Sun and Moon
3rd Bodies
• Right ascension of
the ascending node:

cos i



0
.
00338
Moon
  0.00154 cos i

Sun
n
i = orbit inclination
n = number of orbit revs per day
n
• Argument of perigee


4

5
sin
i
 0.00169
n


4

5
sin
i
 0.00077
2
 Moon
 Sun
2
n
Gravity Gradient, Magnetic Field,
Solar Pressure
3
Tgrav 
I z  I y sin( 2 )
3
2R
Tmag  DB
Fsolar 

c
I = s/c moment of inertia about axis
R = s/c distance from center of Earth
 = angle between Z axis and local vertical
D = s/c electric field strength (Am2)
B = local magnetic field strength (T); varies with R-3
(1   ) A cos 
 = 1367 W/m2 at Earth’s orbit
c = speed of light
= reflectivity
 = angle of incidence
Varying Disturbance Torques
NOTE: The magnitudes of the torques is
dependent on the spacecraft design.
Torque (au)
Drag
Gravity
Solar
Press.
Magnetic
LEO
GEO
Orbital Altitude (au)
Actuators
• Passive
– Gravity Gradient Booms
– Electrodynamic Tethers
• Active
– Magnetic Torque Rods
– Thrusters
Oblate Earth
• The Earth is not a perfect sphere with the mass
at the center (point mass)
– In fact, the Earth has a bulge at the equator and a
flattening at the poles
– Major assumption of the restricted, two-body EOM
• The J2 effects
– RAAN
– Argument of perigee
• Magnitude of the effect is governed by
– Orbital altitude
– Orbital eccentricity
– Orbital inclination
Earth's second-degree zonal spherical harmonic coefficient
J2 Effects
Sun Synchronous Orbit
• Select appropriate inclination of orbit to
achieve a nodal regression rate of ~1º/day
(Orbit 360º in 365 days)
J2 Effects
Molniya Orbit
• Select orbit inclination so that the
argument of perigee regression rate is
essentially zero
– Allows perigee to remain in the hemisphere of
choice
– Allows apogee to remain in the hemisphere of
choice
• VIDEO
J2 Increasing?
J2
Initial decrease thought
to be from a mantle
rebound from melted
ice since the last Ice
Age
Recent increase can
only be caused by a
significant movement of
mass somewhere in
the Earth
C. Cox and B. F. Chao, "Detection of large-scale mass redistribution in the terrestrial system since 1998," Science, vol 297, pp 831, 2 August 2002.