Appendix : Rigorous Derivation of Realizable Launch Inclination • Launch Initial Conditions North Pole Greenwich Meridian • Position: λ, Latitude z Ω, Longitude (Inertial) Launch meridian h, Altitude Equator y Re δ x Vernal Equinox θg 1 MAE 5540 - Propulsion Systems What Happens at Launch? 2 MAE 5540 - Propulsion Systems What Happens at Launch? • Velocity and Position at Burnout Determine Orbit Final Stage Burnout 3 MAE 5540 - Propulsion Systems What HappensWhat happens at Launch on Launch? • Example Kennedy Space Center (KSC) Due-East Launch 28.5° Inclination Orbit 28.5° 4 MAE 5540 - Propulsion Systems Launch? …" 5 MAE 5540 - Propulsion Systems Launch? 6 MAE 5540 - Propulsion Systems What Happens on Launch? (cont'd) • ExampleLaunch? Kennedy Space Center (KSC) 30° South (from east)Launch Direction 40.44° Orbit Inclination Huh? 40.44° 28.5° 7 MAE 5540 - Propulsion Systems Achieved Orbit Inclinations 0° Launch Angle 28.6° Inclination -90° Launch Angle 90° Inclination = 30° Launch Angle 40.44° Inclination -30° Launch Angle 40.44° Inclination What?! Doesn't add up ... 8 MAE 5540 - Propulsion Systems "Spherical Geometry" KennedySpherical Geometry Space Center (KSC) 30° North (from east)Launch Direction • Non-Euclidian Space ... 40.44° Orbit Inclination Addition Doesn't Exist! in this Abstract Space φ −− launch angle φ λ −− launch latitude i −− inclination angle "Spherical Triangle" λ i 9 MAE 5540 - Propulsion Systems "Spherical Geometry" Spherical …" “fixed earth • Non-Euclidian Space ... Addition Doesn't approximation” Exist! in this Abstract Space cos i = cos φ cos λ φ −− launch angle λ −− launch latitude i inclination angle Launch Angle sometimes expressed as −− "azimuth" ... angle from local true north 90° - φ ⇒ degrees Az = π - φ ⇒ radians 2 • Goingcos (toi )have= ctoos push(λ the)⋅ s"faithin(Az) • then a miracle occurs: Meter" pretty hard here 10 MAE 5540 - Propulsion Systems Direct Launch Inclination Angle 180° i = cos-1 cos 30° π × cos 28.6° π = 40.44° π 180° 180° 180° i = cos-1 cos -30° π × cos 28.6° π = 40.44 π 180° 180° 11 MAE 5540 - Propulsion Systems Achievable Direct-Launch Achievable Direct Launch Inclination Angles Inclination Angles 160 Orbit Inclination Angle, ° • No way to get directly 140 to Equatorial orbit from KSC ° 120 •28.6° inclination is lower limit 100 Retrograde Orbit 80 60 Orbit Inclination, Polar orbit 40 KSC Launch Latitude, 28.6° 20 -200 -100 0 100 200 Launch Angle (Counterclockwise ° from due east) 12 MAE 5540 - Propulsion Systems Achievable Direct-Launch InclinationAchievable … Angles 160 Orbit Inclination Angle, ° ° Almost can Achieve 120 Direct launch to Equatorial orbit 80 Retrograde Orbit 40 Orbit Inclination, French Guyana (Ariane) Polar orbit Launch Latitude, 5.3° 0 -200 -100 0 100 200 Launch Angle (Counterclockwise ° from due east) 13 MAE 5540 - Propulsion Systems Bottom Line Bottom Line See following slides for rigorous calculation • Physically Impossible to Launch Directly into an orbit with a Lower inclination Angle than the Launch latitude •Physically Possible to launch directly into any orbit with an inclination angle greater than or equal to launch latitude 14 MAE 5540 - Propulsion Systems • Launch Initial Conditions North Pole Greenwich Meridian z Launch meridian • Launch Initial Conditions (Inertial Coordinates) Equator y Re δ x Vernal Equinox θg Sidereal hour angle 15 MAE 5540 - Propulsion Systems Sidereal Hour Angle Ignore for now! 16 MAE 5540 - Propulsion Systems Computing the Hour Angle 17 MAE 5540 - Propulsion Systems • Earth Radius • Earth radius as Function of Latitude 18 MAE 5540 - Propulsion Systems • Initial Velocity Vector" after the rocket “turns the corner” V0 • Launch Initial Conditions • Velocity: Vo (earth relative vel.) γ (flight path angle) φ (launch azimuth) 19 MAE 5540 - Propulsion Systems • Initial Velocity Vector" after the rocket “turns the corner” • Launch Initial Conditions North Pole Greenwich Meridian z Launch meridian Equator y Re + rotational velocity of earth δ x Vernal Equinox θg 20 MAE 5540 - Propulsion Systems • Earth “Boost” Earth Rotates Under the Orbit • Launch Initial Conditions Fixed in (earth) Inertial Space V boost acts due east V"boost" 21 MAE 5540 - Propulsion Systems • Angular Velocity of Earth Earth Rotates Under the Orbit • Launch Initial Conditions Fixed in (earth) Inertial Space V boost acts due east V"boost" 22 MAE 5540 - Propulsion Systems • Initial Velocity Vector" after the rocket “turns the corner” • Launch Initial Conditions 23 MAE 5540 - Propulsion Systems Inertial Velocity (Initial condition) Rotate Through Latitude First 2-rotation No 1 (X-axis) rotation Needed Why? z` already points Towards center of earth 24 MAE 5540 - Propulsion Systems Inertial Velocity (Initial condition) Rotate Through Longitude Next 25 MAE 5540 - Propulsion Systems Inertial Velocity (Initial condition) 26 MAE 5540 - Propulsion Systems Initial Conditions In Inertial Coordinates 27 MAE 5540 - Propulsion Systems Out-of-plane orbital elements (cont’d) l =R x V 28 MAE 5540 - Propulsion Systems 29 MAE 5540 - Propulsion Systems 30 MAE 5540 - Propulsion Systems 31 MAE 5540 - Propulsion Systems 32 MAE 5540 - Propulsion Systems 33 MAE 5540 - Propulsion Systems .