Also S=sIny 0 So that rs=Rg(cos Cose siny Cosy sine) r×s=R。 cos y siny sine sin y Cos0-CoS y Cosy sine The earth does not advance much in its orbit in one day. Also, of course, Y In performing the averaging of Equation (3), we keep y(approximately) const constant. When(6)and(7)are multiplied together we obtain sine and cose quadratic combinations, and we use 1 sine cose de=0 1 sin e de case de 2兀 and therefore the mean torque is RG siny sin v cosy CoS y sIn y COS This torque will precess the geosynchronous orbit by changing its specific angular momentum (=RGOG nG(ng=unit vector along the Geographical North direction, OG 86400ads) le are interested in the long-term(secular)effects. For times much longer than year, we can also average the torque over the angle y. Noticing that, as before (sin y COs v)=0, -RG siny COS 0 16.522, Space P pessan Lecture 4 Prof. Manuel martinez Page 6 of 10Also ⎧cos ψ⎫ G ⎪ s= ⎨sin ψ ⎪ ⎬ (5) ⎪ ⎩0 ⎪ ⎭ So that G G r.s = RG (cos ψ cos θ + sin ψ cos sin γ θ) (6) ⎧-sin ψ sin γ sin θ ⎫ G G ⎪ ⎪ r s=R × G ⎨cos ψ sin γ sin θ ⎬ (7) ⎪ ⎩sin ψ cos θ - cos ψ cos sin θ⎪ γ ⎭ In performing the averaging of Equation (3), we keep ψ (approximately) constant. The Earth does not advance much in its orbit in one day. Also, of course, γ is constant. When (6) and (7) are multiplied together we obtainsin θ and cos θ quadratic combinations, and we use 1 2π 1 2π 2π 1 d d d 2π ∫0 sinθ cosθ θ=0 ; 2π ∫0 sin2 θ θ= 2 1 π ∫0 cos2 θ θ= (8) 2 and therefore the mean torque is JG 3 µ ⎧- cos γ sin ψ⎫ q = 3 S 2 ⎪ ⎪ RG sin γ sin ψ ⎨cos γ cos ψ ⎬ (9) ⎪ 2 RES ⎩sin γ cos ψ ⎪ ⎭ This torque will precess the geosynchronous orbit by changing its specific angular G G G momentum 2 A = RG ω n (nG =unit vector along the Geographical North G G direction, ωG = 2π rad s ). 86400 We are interested in the long-term (secular) effects. For times much longer than 1 2year, we can also average the torque over the angle ψ . Noticing that, as before, sin ψ cos ψ = 0 , sin2 ψ = 1, 2 ⎧ ⎫ 1 JG 3 µs 2 ⎪ ⎪ q = − R sin γ cos γ ⎨ ⎬ 0 (10) sec ular 4 R3 G ES ⎪ ⎪ ⎩ ⎭ 0 16.522, Space Propulsion Lecture 4 Prof. Manuel Martinez-Sanchez Page 6 of 10