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Orbital Mechanics II: Transfers, Rendezvous, Patched Conics, and Perturbations Dr. Andrew Ketsdever Lesson 3 MAE 5595

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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)

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Hohmann Transfer

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2 2 V1V1 ΔV1ΔV1 Conceptual Walkthrough alt 1 = 300 km alt 2 = 1000 km Slides Courtesy of Major David French, USAFA/DFAS

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2 2 V t1

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2 2 V t2 ΔV2ΔV2

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2 2 V2V2

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2 2 Time of Flight

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Hohmann Transfer

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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

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One-Tangent Burn

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Spiral Transfer Expect to multiply by as much as a factor of 2 for some missions

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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

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Plane Changes Simple – Combined –

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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)

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Co-Orbital Rendezvous

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Target Leading

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Co-Orbital RendezvousTarget Leading

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Co-Orbital RendezvousTarget Leading 3 step process for determining phasing orbit size

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Co-Orbital RendezvousTarget Leading ω TGT 1

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Co-Orbital RendezvousTarget Leading Φ travel ω TGT 2

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Co-Orbital RendezvousTarget Leading Φ travel ω TGT 3

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Co-Orbital RendezvousTarget Trailing

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Co-Orbital RendezvousTarget Trailing

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Co-Orbital RendezvousTarget Trailing ω TG T Φ travel

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Co-Planar Rendezvous

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Coplanar Rendezvous

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

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2 2 ω INT ω TGT 1

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2 2 TOF 2

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2 2 α lead ω INT ω TGT TOF 3

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2 2 α lead Φ final ω INT ω TGT 4

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2 2 α lead Φ final ω INT ω TGT 5 Φ initial

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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)

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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

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Patched Conic Approximation Geo: Hyperbolic escape Helio: Elliptical transfer Targeto: Hyperbolic arrival

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Several factors cause perturbations to a spacecraft’s attitude and/or orbit –Drag –Earth’s oblateness –Actuators –3 rd bodies –Gravity gradient –Magnetic fields –Solar pressure Orbital Perturbations

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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

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

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

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Gravity Gradient, Magnetic Field, Solar Pressure = 1367 W/m 2 at Earth’s orbit c = speed of light = reflectivity = angle of incidence 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 (Am 2 ) B = local magnetic field strength (T); varies with R -3

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Varying Disturbance Torques Orbital Altitude (au) Torque (au) Solar Press. Drag Gravity Magnetic LEOGEO NOTE: The magnitudes of the torques is dependent on the spacecraft design.

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Actuators Passive –Gravity Gradient Booms –Electrodynamic Tethers Active –Magnetic Torque Rods –Thrusters

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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

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J2 Effects

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Sun Synchronous Orbit Select appropriate inclination of orbit to achieve a nodal regression rate of ~1º/day (Orbit 360º in 365 days)

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J2 Effects

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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

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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.Science,

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