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
22 V1
ΔV1
Conceptual Walkthroughalt1 = 300 kmalt2 = 1000 km
Slides Courtesy of Major David French, USAFA/DFAS
22Vt2
ΔV2
22V2
22
2
P 3
transtrans a
TOF Time of Flight
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
Φtravel
ωTGT
2Tgt
travelTOF
Co-Orbital Rendezvous Target Leading
Φtravel
ωTGT
3
3
2 phaseaTOF
Co-Orbital Rendezvous Target Trailing
Co-Orbital Rendezvous Target Trailing
Co-Orbital Rendezvous Target TrailingωTGT
Φtravel
Co-Planar Rendezvous
Coplanar Rendezvous
22
5 step process for determining wait time (WT)
22
ωINT
ωTGT
1
22
TOF
2
3transfera
TOF
22
αlead ωINT
ωTGT TOF
3
TOFettlead arg
22
αlead
Φfinal
ωINT
ωTGT
4
leadfinal
22
αlead
Φfinal
ωINT
ωTGT
5
Φinitial
erceptett
initialfinalWTintarg
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
• 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 Perturbations
Orbital Drag
FrontalDDrag AVCF 2
2
1
• 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
3rd Bodies
• Geosynchronous Equatorial Orbits are influenced by the Sun and Moon
3rd Bodies
• Right ascension of the ascending node:
• Argument of perigee
nini
Sun
Moon
cos00154.0
cos00338.0
ni
ni
Sun
Moon
2
2
sin5400077.0
sin5400169.0
i = orbit inclinationn = number of orbit revs per day
Gravity Gradient, Magnetic Field, Solar Pressure
)2sin(2
33
yzgrav II
RT
cos)1( A
cFsolar
= 1367 W/m2 at Earth’s orbitc = speed of light= reflectivity = angle of incidence
I = s/c moment of inertia about axisR = s/c distance from center of Earth = angle between Z axis and local vertical
DBTmag D = s/c electric field strength (Am2)B = local magnetic field strength (T); varies with R-3
Varying Disturbance Torques
Orbital Altitude (au)
Tor
que
(au)
SolarPress.
Drag
Gravity
Magnetic
LEO GEO
NOTE: The magnitudes of the torques isdependent on the spacecraft design.
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?
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
J2
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.