Orbits: Select, Achieve, Determine, Change Physical Background Newton, Kepler et al. Coordinate systems Orbit Transfers Orbit Elements Orbit survey LEO / MEO / GTO / GEO Sun Synch Interplanetary, escapes Capture, flyby & assist Ask an Orbitalogist if you need to: Stay solar illuminated Overfly @ constant time of day Maintain constant position (over equator, pole, sun/earth) With another satellite Constellation configuration Rendezvous Escape / assist / capture Determine orbit from observation Determine location from orbit Optimize Ground Station location Estimate orbit lifetime + tell you nav strategy & ∆V Engin 176 Meeting #5

(Re) Orientation 1 - Introduction 2 - Propulsion & ∆V 3 - Attitude Control & instruments 4 - Orbits & Orbit Determination LEO, MEO, GTO, GEO Special LEO orbits Orbit Transfer Getting to Orbit GPS 5 - Launch Vehicles 6 - Power & Mechanisms 7 - Radio & Comms 8 - Thermal / Mechanical Design. FEA 9 - Reliability 10 - Digital & Software 11 - Project Management Cost / Schedule 12 - Getting Designs Done 13 - Design Presentations Another night of F=MA? Education is the process of realizing that you don’t know what you thought you knew... Engin 176 Meeting #5

Review of Last time Attitude Determination & Control Design Activity Feedback Control Systems description Simple simulation Attitude Strategies The simple life Eight other approaches and variations Disturbance and Control forces (note re CD>1) Design build & test an Attitude Control System Design Activity Team designations Mission selections Homework - ACS for mission Or in a word, F=MA Plant (satellite) Set point Error Control Algorithm Sensor Disturbances Actuator Engin 176 Meeting #5

Assignments for February 21 Orbits Select minimum 2, preferable 3 orbits your mission could use Create a trade table comparing them Criteria could include: Mission suitability (e.g. close or far enough) Revisit or other attributes Cost to get there - and stay there Environment for spacecraft For the selected orbit Describe it (some set of orbit elements) How will you get there? How will you stay there? Estimate radiation & drag v b v r X F F’ c rp ra Reading SMAD 18 SMAD 17 (if you haven’t already) TLOM: Launch sites Or in a word, F=MA a Engin 176 Meeting #5

LEO vs. GEO Orbit LEO: 1000 km GEO: 36,000 km Low launch cost/risk Short range Global coverage (not real time) Easy thermal environment Magnetic ACS Multiple small satellites / financial “chunks” Minimal propulsion GEO: 36,000 km Fixed GS Antenna Constant visibility from 1 satellite Nearly constant sunlight Zero doppler Engin 176 Meeting #5

Describing Orbits Kepler’s first law: All orbits are described by a Conic Section. rF + rF’ = constant Defines ellipse, circle, parabola, hyperbola rF’ rF F F’ Engin 176 Meeting #5

Elliptical Orbit Parameters r’’ = G(M +m)/r2 (r/r) = -µ/ r2 (r/r) Two-Body equation v V (true anomaly) b r X F F’ c rp ra a Engin 176 Meeting #5

Circles, Ellipses and Beyond Planets, Moons, LEOs, GEOs, Vcirc = [µ/r]1/2 Vesc =[2]1/2 Vcirc T = 2π (a3/µ)1/2 (kepler’s 3rd Law) Orbit elements: r, i (T, i) plus tp, q0… (epoch) Hyperbolic Asymptote r b v rp a Note to Orbital Racers: Lower means: - Higher velocity and - Shorter Orbit Period Engin 176 Meeting #5

Circles, Ellipses and Beyond Transfer, Molniya, Reconnaissance orbits Comets, Asteroids Real Planets, Moons, LEOs, GEOs Kepler’s 2nd law e = c / a r = p / [1+ e cos(v)] Hyperbolic Asymptote r b v rp a c Orbit Elements: a (or p), e (geometry) plus i p= a(1-e2) Ω (longitude of ascending node) w (argument of periapsis, ccw from Ω) tp, q0… (epoch) Engin 176 Meeting #5

Circles, Ellipses and Beyond Parabola: (mostly synthetic objects) Escape (to V∞= 0) V(parabola) = Vesc = [2p/r]1/2 Hyperbolic Asymptote r b v rp Hyperbola: (mostly synthetic objects Interplanetary & beyond Escape with V∞> 0 Planetary Assist (accelerate & turn) -> motion of M matters <- e = 1 + V2∞ rp/µ a c Engin 176 Meeting #5

2-Body 2-D Solution NB: 3 terms, a, e, v, completely define position in planar orbit - all that’s left is to define the orientation of that plane r = a (1-e2) / (1 + e cos n) Position, r, depends on: a (semi-major axis) e(eccentricity = c/a (= distance between foci /major axis) n (polar angle or true anomaly) 4 major type of orbits: circle e = 0 a = radius ellipse 0< e < 1 a > 0 parabola e = 1 a = ∞ (eq. above is useless) hyperbola e > 1 a < 0 Engin 176 Meeting #5

The 6 Classical Orbit Elements* 3 elements (previous page) describe the conic section & position. a - semi-major axis - scale ( in kilometers) of the orbit. e - eccentricity - (elliptical, circular, parabolic, hyperbolic) v true anomaly - the angle between the perigee & the position vector to the spacecraft - determines where in the orbit the S/C is at a specific time. 3 additional elements describe the orbit plane itself i - inclination - the angle between the orbit normal and the (earth polar) Z-direction. How the orbit plane is ‘tilted’ with respect to the Equator. Ω - longitude or right ascension of the ascending node - the angle in degrees from the Vernal Equinox (line from the center of the Earth to the Sun on the first day of autumn in the Northern Hemisphere) to the ascending node along the Equator. This determines where the orbital plane intersects the Equator (depends on the time of year and day when launched). w, argument of perigee - the angle in degrees, measured in the direction and plane of the spacecraft’s motion, between the ascending node and the perigee point. This determines where the perigee point is located and therefore how the orbit is rotated in the orbital plane. Engin 176 Meeting #5 *NB: Earth axis rotation is not considered

Why 6 Orbit Elements? 1-D: Example: mass + spring like the dynamic model of last week position (1 number) plus velocity (1 number) necessary 2-D: Example: air hockey puck or single ball on pool table X & Y position, plus Velocity components along X & Y axes 3-D: Example: baseball in flight Altitude and position over field + 3-D velocity vector Alternative Orbit determination systems GPS: Latitude, Longitude, Altitude and 3-D velocity vector Radar: Distance, distance rate, azimuth, elevation, Az rate, El rate Ground sitings: Az El only (but done at many times / locations) Breaking it down: Range R and Velocity V R X V = h angular momentum vector = constant dot prod. with pole to get i e2 = 1 + 2E(h/µ)2 where E = V 2 /2 - µ/r For sing = R . V/RV (g is flight path angle to local horizon): tanq = (RV 2/µ)singcosg / [ (RV 2 /µ)cos 2 g - 1 ] Engin 176 Meeting #5

Orbit facts You Already Know Must be geosynch at the equator (q=0) Orbit planes & inclination are fixed Knowing instantaneous position + velocity fully determines the orbit Orbit plane must include injection point and earth’s CG (hence the concept of a launch window) Dawn / Dusk orbit in June is Noon / Midnight in September Escaping the solar system ¿So how do they do this? Engin 176 Meeting #5

Orbital Trick #1: Orbit Transfer Where new & old orbit intersect, change V to vector appropriate to new orbit If present and desired orbit don’t intersect: Join them via an intermediate that does Do V & i changes where V is minimum (at apogee) Orbit determination: requires a single simultaneous measurement of position + velocity. GPS and / or ground radar can do this. Engin 176 Meeting #5

Orbital Trick #2: Getting There #1: Raise altitude from 0 to 300 km (this is the easy part) Energy = mgh = 100 kg x 9.8 m/s2 x 300,000 m = 2.94 x 108 kg m2/ s2 [=W-s = J] = 82 kw-hr = 2.94 x 106 m2/ s2 per kg ∆V = (E)1/2 = 1715 m/s #2: Accelerate to orbital velocity, 7 km/s (the harder part) ∆V (velocity) = 7000 m/s (80% of V, 94% of energy) ∆V (altitude) = 1715 m/s ∆V (total) = 8715 m/s (+ about 1.5 km/s drag + g loss) Note to Space Tourists: ∆V = gIsp ln(Mo/Mbo) => Mbo / Mo = 1/ exp[∆V/gIsp]) For Isp 420, Mbo = 10% Mo Engin 176 Meeting #5

Orbital Trick #2’: Getting help? Launch From Airplane at 10 km altitude and 200 m/s #1: Raise altitude from 10 to 300 km Energy = mgh = 100kg x 9.8 m/s2 x (300,000 m - 10,000 m) ∆V = (E)1/2 = 1686 m/s (98% of ground based launch ∆V) (or 99% of ground based launch energy) #2: Accelerate to 7 km/s, from 0.2 km/s ∆V (velocity) = 6800 m/s (97% of ground ∆V, 94% of energy) ∆V (∆H) = 1686 m/s (98% of ground ∆V, 96% of energy) ∆V (total, with airplane) = 8486 m/s + 1.3 km/s loss = 8800 m/s ∆V (total, from ground) = 8715 m/s + 1.5 km/s loss = 9200 m/s Velocity saving: 4% Energy saving: 8% Downsides: Human rating, limited dimension & mass, limited propellant choices, cost of airplane (aircraft doesn’t fully replace a stage) Engin 176 Meeting #5

Orbital Trick #3: Sun Synch Earth needs a belt: it is 0.33% bigger (12756 v. 12714 km diameter) in equatorial circumference than polar circumference Earth’s shape as sphere + variations. Potential, U is: U(R, q, f) = -µ/r + B(r, q, f) => U = -(µ/r)[1 - ∑2∞(Re/r)nJnPn (cosq)] Re = earth radius; r = radius vector to spacecraft Jn is “nth zonal harmonic coefficient” Pn is the “nth Legendre Polynomial” J1 = 1 (if there were a J1) J2 = 1.082 x10-3 J3 = -2.54 x10-6 J4 = -1.61 x10-6 Engin 176 Meeting #5

Orbital Trick #3: Sun Synch (continued) Extra Pull Nodal Regression, how it works, and how well Intuitive explanations: #1: Extra Pull causes earlier equator crossing #2: Extra Pull is a torque applied to the H vector Equator Extra Pull Engin 176 Meeting #5

Ordinary Orbits GEO Remote Sensing: LEO Comms: Equatorial: Favors polar, LEO, 2x daily coverage (lower inclinations = more frequent coverage). Harmonic orbit: period x n = 24 (or 24m) hours (n & m integer) LEO Comms: Same! - multiple satellites reduce contact latency. Best if not in same plane. Equatorial: Single satellite provides latency < 100 minutes; minimum radiation environment • Sun Synch: Dawn/Dusk offers Constant thermal environment & constant illumination (but may require ∆V to stay sun synch) Elliptical: Long dwell at apogee, short pass through radiation belts and perigee... Molniya. Low E way to achieve max distance from earth. MEO: Typically 10,000 km. From equator to 45 or more degrees latitude GEO Engin 176 Meeting #5

Lagrange Points Polar Stationary L4 (Stable) L3 (Unstable) Engin 176 Meeting #5

GPS in 1 slide 4 position vectors => 4 pseudo path lengths Solve for 4 unknowns: - 3 position coordinates of user - time correction of user’s clock Freebies - Atomic clock accuracy to user - Velocity via multiple fixes Engin 176 Meeting #5

Non-Obvious Terms Nodes: ascending, descending, line of nodes True Anomaly: angle from perigee Inclination (0, 180, 90, <90, >90) Ascension, Right Ascension Conjunction = same RA (see vernal eq) Argument of perigee (w from RA) Declination (~= elevation) Geoid - geopotential surface Julian Calendar: 365.25 days Gregorian: Julian + skip leap day in 1900, 2100… Ephemerides Frozen Orbits (sun synch, Molniya) Periapsis, Apoapsis Vernal Equinox (equal night) Solstice Ecliptic (and eclipses) Siderial Terminator Azimuth, Elevation Oblate / J2 Term spinning about minor axis (earth) Prolate: spinning about major axis (as a football) Precession: steady variation in h caused by applied torque Nutation: time varying variation in h caused by applied torque Engin 176 Meeting #5