Dr. Andrew Ketsdever Lesson 3 MAE 5595 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 V1 ΔV1 Slides Courtesy of Major David French, USAFA/DFAS

2 2 Vt1

2 2 ΔV2 Vt2

2 2 V2

Time of Flight 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 Combined 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 Co-Planar 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

3 step process for determining phasing orbit size 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 Φtravel 2

Co-Orbital Rendezvous Target Leading ωTGT Φtravel 3

Co-Orbital Rendezvous Target Trailing

Co-Orbital Rendezvous Target Trailing

Co-Orbital Rendezvous Target Trailing ωTGT Φtravel

Co-Planar Rendezvous

Coplanar Rendezvous

5 step process for determining wait time (WT) 2 2 5 step process for determining wait time (WT)

1 2 ωTGT 2 ωINT

2 2 TOF 2

3 2 ωTGT TOF αlead 2 ωINT

4 2 ωTGT Φfinal αlead 2 ωINT

5 2 ωTGT Φfinal αlead 2 ωINT Φinitial

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

3rd Bodies Geosynchronous Equatorial Orbits are influenced by the Sun and Moon

3rd Bodies Right ascension of the ascending node: Argument of perigee i = orbit inclination n = number of orbit revs per day

Gravity Gradient, Magnetic Field, Solar Pressure 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  = 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. Drag Torque (au) Gravity Solar Press. Magnetic LEO GEO Orbital Altitude (au)

Actuators Passive Active Gravity Gradient Booms Electrodynamic Tethers 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.