SUMMER GEOGRAPHIC SIDEREA SOLSTICE 23° S/C IN GEO s TIDAL ong vernal Equinox FORCES Ecliptic, solsticial line z: North sidereal EQUIN If r is the relative position vector of the spacecraft and s is the unit vector from the Sun, the Tidal Force( Gravity gradient) per unit mass is R (1) (This the imbalance between solar attraction and centrifugal force, which cancels on the earth, r=0). Of this, the r part gives no torque about the Earth's center. To compute the effect on the orbit, we smear"the torque r x f about the geosynchronous orbit, and calculate an orbit-average torque(per unit mass) (3) We project onto a set of axis(x, y, z)as defined in the sketch( fixed axis) r=y(=raising cosy here RG is the geosynchronous orbit radius(42, 200 Km)and y=23. 44. is the angle between the equatorial plane and the ecliptic 16.522, Space Propulsion Lecture 4 Prof. Manuel martinez-Sanchez Page 5 of 10θ r r r Z f R ψ S Fi l x y o ~ G.N. TIDAL TORQUE, SUMMER SOLSTICE VERNAL EQUINOX TIDAL FORCES ES GEOGRAPHICAL NORTH SIDEREAL NORTH EARTH xed Axes: x: A ong Verna y: In Ec z: North s SUN 23 /C IN GEO s l Equinox liptic, solsticial line idereal G G If r is the relative position vector of the spacecraft, and s is the unit vector from the Sun, the Tidal Force (Gravity gradient) per unit mass is JG ( GG G G f = µs ⎡3 r.s s − r⎤ ) (1) 3 R ⎣ Es ⎦ (This the imbalance between solar attraction and centrifugal force, which cancels on G G the Earth, r = 0 ). Of this, the r part gives no torque about the Earth’s center. To G G compute the effect on the orbit, we “smear” the torque r f × about the geosynchronous orbit, and calculate an orbit-average torque (per unit mass): JG 1 2π G G q = × 2π ∫0 r f d θ (2) JG 3µ 1 2π G G G G q = 3 s (r.s ) (r × s) dθ (3) REs 2π ∫0 We project onto a set of axis (x, y, z) as defined in the sketch (fixed axis): ⎧ ⎫ x ⎧cos θ ⎫ G ⎪⎪ ⎪ ⎪ r= ⎨ ⎬y = RG ⎨sin θ cos γ⎬ (4) ⎪ ⎪ ⎪ ⎩ ⎭ ⎩sin θ sin γ ⎪ z ⎭ where RG is the geosynchronous orbit radius (42, 200 Km) and γ = 23.44 D is the angle between the equatorial plane and the ecliptic. 16.522, Space Propulsion Lecture 4 Prof. Manuel Martinez-Sanchez Page 5 of 10