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

22 Vt1

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.