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Engineering 176 Orbital Design. Mr. Ken Ramsley kenneth_ramsley@brown.edu (508) 881- 5361. Class Topics. When Orbits Were Perfect (and politically dangerous) Einstein’s Geodesics (the art and science of motion) Kepler’s Three Laws (based on Tycho’s meticulous data)

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Engineering 176Orbital DesignMr. Ken Ramsley kenneth_ramsley@brown.edu (508) 881- 5361Class Topics When Orbits Were Perfect (and politically dangerous) Einstein’s Geodesics (the art and science of motion) Kepler’s Three Laws (based on Tycho’s meticulous data) Orbital Elements Defined and Illustrated Useful Orbits and Maneuvers to Get There Interplanetary Space and BeyondEN176 Orbital DesignThe AncientsAristotle (384 BC – 322 BC)Claudius Ptolemaeus (AD 83 – c.168)Copernicus and TychoNicolaus Copernicus (1473 - 1543)Tycho Brahe (1546 - 1601)The Copernicus Solar SystemImage: Courtesy of tychobrahe.comTycho Brahe's UraniborgObservatory and 90° Star Sighting QuadrantKepler and GalileoJohannes Kepler (1571 - 1630)Galileo Galilei (1564 - 1642)Newtonand LaGrangeIsaac Newton (1643 - 1727)Joseph Louis Lagrange (1736-1813)EinsteinAll objects in motion conserve momentum through a balance ofGravity Potential andVelocity Vector(think rollercoaster)Geodesics:The Science and Art of 4D Curved Space Trajectories.Defining Simple 2-Body OrbitsThis is all we need to know…Shape – More like a circle, or stretched out? Size – Mostly nearby, or farther into space? OrbitalPlaneOrientation– Pitch,Yaw,andRoll Satellite Location – Where are we in this orbit? Kepler’s First LawEvery orbit is an ellipse with the Sun (main body) located at one foci.Kepler’s Second LawDay 40Day 30Day 50Day 60Day 20Day 70Day 80Day 90Day 10Day 100Day 110Day 0A line between an orbiting body and primary body sweeps out equal areas in equal intervals of time.Day 120Kepler’s Third LawThis defines the relationship of Orbital Period & Average Radius for any two bodies in orbit.For a given body, the orbital period and average distance for the second orbiting body is:P2 = R3EXAMPLE:EarthP = 1 YearR = 1 AUMarsP = 1.88 YearsR = 1.52 AUR2R1P1P2P = Orbital PeriodR = Average RadiusVernal Equinox – The Celestial BaselineFirst some astronomy…June 21stWhen the Sun passes over the equator moving south to north.Vernal Equinox(March 20th) Defines a fixed vector in space through the center of the Earth to a known celestial coordinate point.SunEpoch 2000The Vernal Equinox drifts ~0.014° / year. Orbits are therefore calculated for a specified date and time, (most often Jan 1, 2000, 2050 or today).December 22ndConic Sections(shape) Eccentricitye=0 -- circle e<1 -- ellipse e=1 -- parabola e>1 -- hyperbola e < 1 Orbit is ‘closed’ – recurring path (elliptical) e > 1 Not an orbit – passing trajectory (hyperbolic)Keplerian Elementse,a,andv (3 of 6)e120°150°90°Eccentricity(0.0 to 1.0)vTrue anomaly (angle)aApogee 180°Perigee 0°Semi-major axis (nm or km)e=0.8 vrse=0.0e defines ellipse shape a defines ellipse size v defines satellite angle from perigeeApo/Peri gee– Earth Apo/Peri lune – Moon Apo/Peri helion–Sun Apo/Peri apsis– non-specificInclinationi(4th Keplerian Element)Intersection of the equatorial and orbital planesiInclination (angle)(above)(below)Ascending NodeEquatorial Plane ( defined by Earth’s equator )Sample inclinations0° -- Geostationary 52° -- ISS 98° -- MappingAscending Node is where a satellite crosses the equatorial plane moving south to northRight Ascension[1] of the ascending nodeΩandArgument of perigeeω(5th and 6th Elements)Ω = angle from vernal equinox to ascending node on the equatorial planePerigee Directionω = angle from ascending node to perigee on the orbital planeωΩAscending Node[1]Right Ascension is the astronomical term for celestial (star) longitude.Vernal EquinoxThe Six Keplerian Elementsa=Semi-major axis (usually in kilometers or nautical miles)e=Eccentricity (of the elliptical orbit)v=True anomalyThe angle between perigee and satellite in the orbital plane at a specific timei=InclinationThe angle between the orbital and equatorial planes Ω=Right Ascension (longitude) of the ascending nodeThe angle from the Vernal Equinox vector to the ascending node on the equatorial planew=Argument of perigee The angle measured between the ascending node and perigeeShape, Size, Orientation, and Satellite Location.Sample Keplerian Elements(ISS)TWO LINE MEAN ELEMENT SET - ISS 1 25544U 98067A 09061.52440963 .00010596 00000-0 82463-4 0 9009 2 25544 51.6398 133.2909 0009235 79.9705 280.2498 15.71202711 29176 Satellite: ISS Catalog Number: 25544 Epoch time: 09061.52440963 = yrday.fracday Element set: 900 Inclination: 51.6398 degRA of ascending node: 133.2909 degEccentricity: .0009235Arg of perigee: 79.9705 degMean anomaly: 280.2498 degMean motion: 15.71202711 rev/day(semi-major axis derivable from this)Decay rate: 1.05960E-04 rev/day^2 Epoch rev: 2917 Checksum: 315State VectorsNonKeplerian Coordinate SystemCartesian x, y, z, and 3D velocityOrbit determinationOn Board GPS Ground Based Radar:Distance or “Range” (kilometers). Elevation or “Altitude” (Horizon = 0°, Zenith = 90°). Azimuth (Clockwise in degrees with due north = 0°).On board Radio TransponderRanging:Alt-Az plus radio signal turnaround delay (like radar).Ground Sightings:Alt-Az only (best fit from many observations).Launch From Vertical TakeoffRaising your altitude from 0 to 300 km(‘standing’ jump) Energy= mgh = 1 kg x 9.8 m/s2 x 300,000 m ∆V =1715 m/s 7 km/s lateral velocity at 300 km altitude(orbital insertion) ∆V (velocity) = 7000 m/s ∆V (altitude) = 1715 m/s ∆V (total) = 8715 m/s[1] [1] plus another 1500 m/s lost to drag during early portion of flight.Launch From Airplane at 200 m/sand10 km altitudeRaise altitude from 10 to 300 km (‘flying’ jump)Energy= mgh = 1 kg x 9.8 m/s2 x 290,000 m ∆V = 1686 m/s(98% of ground based launch ∆V)(96% of ground based launch energy)Accelerate to 7000 m/s from 200 m/s ∆V (velocity) = 6800 m/s (97% of ground ∆V, 94% of energy) ∆V (∆Height)= 1686 m/s (98% of ground ∆V, 96% of energy) ∆V (total, with airplane) = 8486 m/s + 1.3 km/s drag loss = 9800 m/s ∆V (total, from ground) = 8715 m/s + 1.5 km/s drag loss = 10200 m/sTotal Velocity savings: 4%, Total Energy savings: 8%Downsides: Human rating required for entire system, limited launch vehicle dimension and mass, fewer propellant choices, airplane expenses.Ground TracksGround tracks drift westward as the Earth rotates below an orbit. Each orbit type has a signature ground tract.More Astronomy FactsThe SunDrifts east in the sky ~1° per day. Rises 0.066 hours later each day. (because the earth is orbiting)The Earth…Rotates 360° in 23.934 hours (Celestial or “Sidereal” Day)Rotates ~361° in 24.000 hours (Noon to Noon or “Solar” Day)Satellites orbits are aligned to the Sidereal day – not the solar dayOrbital Perturbations“All orbits evolve”Atmospheric Drag (at LEO altitudes, only) – Worse during increased solar activity. – Insignificant above ~800km.Nodal Regression – The Earth is an oblate spheroid. This adds extra “pull” when a satellite passes over the equator – rotating the plane of the orbit to the east.Other Factors– Gravitational irregularities – such as Earth-axis wobbles, Moon, Sun, Jupiter gravity (tends to flatten inclination). Solar photon pressure. Insignificant for LEO – primary perturbations elsewhere.‘LEO’ < ~1,000km (Satellite Telephones, ISS)‘MEO’ = ~1,000km to 36,000km (GPS)‘GEO’ = 36,000km (CommSats, HDTV)‘Deep Space’ > ~GEOLEO is most common, shortest life. MEO difficult due to radiation belts. Most GEO orbit perturbation is latitude drift due to Sun and Moon.Nodal RegressionOrbital planes rotate eastward over time.(above)Ascending Node(below)Nodal Regression can be very useful.Sun-Synchronous OrbitsRelies on nodal regression to shift the ascending node ~1° per day.Scans the same path under the same lighting conditions each day.The number of orbits per 24 hours must be an even integer (usually 15).Requires a slightly retrograde orbit (I = 97.56° for a 550km / 15-orbit SSO).Each subsequent pass is 24° farther west (if 15 orbits per day).Repeats the pattern on the 16th orbit (or fewer for higher altitude SSOs).Used for reconnaissance (or terrain mapping – with a bit of drift).Molniya - 12hr Period‘Long loitering’ high latitude apogee. Once used used for early warning by both USA and USSR‘Tundra’ Orbit - 24hr PeriodHigher apogee than Molniya. For dwelling over a specific upper latitude (Used only by Sirius)GPS Constellation ~ 20200kmalt.GPS: Six orbits with six equally-spaced satellites occupying each orbit.Hohmann Transfer OrbitHohmann transfer orbit intersects both orbits.Requires co-planar initial and ending orbits.After 180°, second burn establishes the new orbit.Can be used to reduce or increase orbit altitudes.By far the most common orbital maneuver.Orbital Plane ChangesBurn must take place where the initial and target planes intersect.Even a small amount of plane change requires lots of ΔV Less ΔV required at higher altitudes (e.g., slower orbital velocities).Often combined with Hohmann transfer or rendezvous maneuver.θSimple Plane Change Formula (No Hohmann component):Plane Change ΔV = 2 x Vorbit x sin(θ/2)Example: Orbit Velocity = 7000m/s, Target Inclination Change = 30°Plane Change ΔV = 2 x 7000m/s x sin(30°/ 2)Plane Change ΔV = 3623m/sFast Transfer OrbitRequires less time due to higher energy transfer orbit.Also faster since transfer is complete in less 180°.Can be used to reduce or increase orbit altitudes.Less common than Hohmann Typically an upper stage restart where excess fuel is often available.Geostationary Transfer Orbit ‘GTO’Requires plane change and circularizing burns.Less plane changing is required when launched from near the equator.2. Plane change where GTO plane intersects GEO plane1. launch to ‘GTO’3. Hohmann circularizing burn‘Super GTO’3. Second Hohmann burn circularizes at GEOGEO Target OrbitInitial orbit has greater apogee than standard GTO.Plane change at much higher altitude requires far less ΔV.PRO: Less overall ΔV from higher inclination launch sites.CON: Takes longer to establish the final orbit.1. Launch to ‘Super GTO’2. Plane change plus initial Hohmann burnLow Thrust Orbit TransferA series of plane and altitude changes.Continuous electric engine propulsion.PROs: Lower mass propulsion system.Same system used for orbital maintenance. CONs: Weeks or even months to reach final orbit. Van Allen Radiation belts.RendezvousLaunch when the orbital plane of the target vehicle crosses launch pad.(Ideally) launch as the target vehicle passes straight overhead.Smaller transfer orbits slowly overtake target (because of shorter orbit periods).Course maneuvers designed to arrive in the same orbit at the same true anomaly.Apollo LM and CSM RendezvousOrbital Debrisa.k.a., ‘Space Junk’February 2009 Iriduim / Cosmos collision created > 1,000 items > 10cm diameterCurrently > 19,000 items 10cm or larger. ~ 700 (4%) functioning S/C. In as few as 50 years, upper LEO and lower MEO may be unusable.Deep SpaceCassini – Saturn orbit insertion using good ‘ol fashion rocket power.Using Lagrange Points to ‘stay put’Halo Orbits(stability from motion)AeroBrakingEarth, Mars, Jupiter, etc.“The poor man’s Hohmann maneuver”The Solar System ‘Super Highway’…designing geodesic trajectories – like tossing a message bottle into the sea at exactly the right time, direction, and velocity.Gravity Assist(Removing Velocity)Gravity Assist (adding velocity)Solar EscapeMultiple Mission TrajectoriesComplex Orbital TrajectoriesGalileo (Jupiter)Cassini (Saturn)Designing Deep Space Missions…yes, there are software tools for thisAssignments for April 2Reading on Orbits:SMAD ch 6 – scan 5 and 7TLOM ch 3 and 4 – scan 5 and 17HOMEWORK: Design minimum two, preferably three orbits your mission could use.For the selected orbits:Describe it (orbital elements) How will you get there? How will you stay there? Estimate perturbationsCreate a trade table to compare orbit designs.Trade criteria should include: Orbit suitability for mission. Cost to get there – and stay there. Space environment (e.g., radiation).Engineering 176 Orbits

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