Cheat Sheet

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ORBITAL MECHANICS ^ − 𝜇 = 𝐺 𝑚𝑝 𝜀 = 𝑣2 2 − 𝜇 𝑟 = − 𝜇 2𝑎 𝑎 = 𝑟𝑝 + 𝑟𝑎 2 = ℎ2 𝜇 1 1 − 𝑒2 𝑏 = 𝑎√1 − 𝑒2 𝑝 = ℎ2 𝜇 = 𝑎 (1 − 𝑒2) 𝑝 = 2𝑟𝑝𝑟𝑎 𝑟𝑝 + 𝑟𝑎 𝒉 = 𝒓 × 𝒗 ℎ = 𝑟2𝜗̇ = 𝑟𝑣𝜗 = √𝑝𝜇 ℎ = √𝑟𝑝𝜇(1 + 𝑒) 𝑑𝐴 𝑑𝑡 = ℎ 2 𝒆 = 𝒗 × 𝒉 𝜇 − 𝒓 𝑟 𝑒 = 𝑟𝑎 − 𝑟𝑝 𝑟𝑎 + 𝑟𝑝 𝑒 = √ℎ2 𝜇2 𝑣𝑟2 + ( ℎ2 𝜇 𝑟 − 1) 2 𝑟𝑝 = ℎ2 𝜇 1 1 + 𝑒 = 𝑎(1 − 𝑒) 𝑟𝑎 = ℎ2 𝜇 1 1 − 𝑒 = 𝑎(1 + 𝑒) 𝑣𝑟 = 𝜇 ℎ 𝑒 𝑠𝑖𝑛𝜗 𝑣𝜗 = 𝜇 ℎ (1 + 𝑒 𝑐𝑜𝑠𝜗) 𝑣𝑝 = √ 𝜇 𝑝(1 + 𝑒) 𝑣𝑎 = √ 𝜇 𝑝(1 − 𝑒) 𝑣𝑝 = √2𝜇 ( 1 𝑟𝑝 − 1 𝑟𝑝 + 𝑟𝑎 ) 𝑣𝑎 = √2𝜇 ( 1 𝑟𝑎 − 1 𝑟𝑝 + 𝑟𝑎 ) tan 𝛾 = 𝑣𝑟 𝑣𝜗 = 𝑒 𝑠𝑖𝑛𝜗 1 + 𝑒 𝑐𝑜𝑠𝜗 𝑇 = 2𝜋√𝑎3 𝜇 ELLIPTICAL ORBITS 0 < 𝑒 < 1 ; 𝜀 < 0 𝐴𝑒𝑙𝑙𝑖𝑝𝑠𝑒 = 𝜋𝑎𝑏 CIRCULAR ORBITS 𝑒 = 0 ; 𝜀 < 0 𝑟 = ℎ2 𝜇 = 𝑎 𝑣 = 𝑣𝜗 = √𝜇 𝑟 ℎ = √𝜇𝑟𝑐 = 𝑟𝑣 PARABOLIC ORBITS 𝑒 = 1 ; 𝜀 = 0 𝑣𝑝 = √2𝜇 𝑟 = 𝑣𝑒𝑠𝑐 𝑟𝑝 = 𝑝 2 tan 𝛾 = tan 𝜗 2 HYPERBOLIC ORBITS cos 𝜗∞ = − 1 𝑒 sin 𝜗∞ = 1 𝑒 √𝑒2 − 1 𝛿 = 2 asin 1 𝑒 𝑣∞ = √− 𝜇 𝑎 𝑣𝑝2 2 − 𝜇 𝑟𝑝 = 𝑣∞2 2 = − 𝜇 2𝑎 ∆= −𝑖𝑏 = −𝑎√𝑒2 − 1 ∆ = − 𝑎 tan 𝛿 2 = 𝑟𝑝𝑣𝑝 𝑣∞ = ℎ 𝑣∞ 𝐶3 = 𝑣∞2 = 𝑣2 − 𝑣𝑒𝑠𝑐2 𝑒 = 1 + 𝑟𝑝𝑣∞2 𝜇 = − 𝑟𝑝 𝑎 + 1 𝑒 = − ∆ + 𝑟𝑝 ∆ − 𝑟𝑝 ℎ = 𝑟𝑝√𝑣∞2 + 2𝜇/𝑟𝑝 = 𝜇/𝑣∞√𝑒2 − 1 TIME LAWS ELLIPTICAL ORBIT 𝐸 − 𝑒 sin 𝐸 = √ 𝜇 𝑎3 (𝑡 − 𝑡0) tan 𝜗 2 = √1 + 𝑒 1 − 𝑒 tan 𝐸 2 PARABOLIC ORBIT √ 𝜇 𝑎3 (𝑡 − 𝑡𝑃)= 1 2 (𝐷 + 𝐷3 3 ) tan 𝜗 2 = 𝐷 HYPERBOLIC ORBIT √ 𝜇 𝑎3 (𝑡 − 𝑡𝑃)= 𝑒 sinh 𝐹 − 𝐹 tan 𝜗 2 = √1 + 𝑒 1 − 𝑒 tanh 𝐹 2 GROUND TRACK ∆𝜆 = 𝑇 𝜔𝐸 𝜔𝐸 = 15.4° ECEI FRAME 𝒓 = [𝑟𝑥 𝑟𝑦 𝑟𝑧] 𝒓 = 𝑟 [ cos 𝛿 cos 𝛼 cos 𝛿 sin 𝛼 sin 𝛿 ] 𝛼 = atan 𝑟𝑦 𝑟𝑥 𝛼 = { acos 𝑟𝑥 𝑟 𝑖𝑓 𝑟𝑦 ≥ 0 2𝜋 − acos 𝑟𝑥 𝑟 𝑖𝑓 𝑟𝑦 < 0 𝛿 = asin 𝑟𝑧 𝑟 PERIFOCAL FRAME 𝑟𝑃𝐹 = 𝑟 ( cos 𝜗 sin 𝜗 0 ) 𝑣𝑃𝐹 = √ 𝜇 𝑝 ( −sin 𝜗 𝑒 + cos 𝜗 0 ) FRAME TRANSFORMATION ECEI = 𝑅3(𝜔)𝑅1(𝑖)𝑅3(𝛺) 𝑃𝐹 𝑃𝐹 = 𝑅3 𝑇(𝛺)𝑅1 𝑇(𝑖)𝑅3 𝑇(𝜔)ECEI COSINE LAW cos 𝑎 = cos 𝑏 cos 𝑐 + sin 𝑏 sin 𝑐 cos 𝐴 cos 𝐴 = −cos 𝐵 cos 𝐶 + sin 𝐵 sin 𝐶 cos 𝑎 SINE LAW sin 𝑏 sin 𝐵 = sin 𝑐 sin 𝐶 = sin 𝑎 sin 𝐴 COTANGENT LAW ctan 𝑐 sin 𝑎 = cos 𝑎 cos 𝐵 + sin 𝐵 ctan 𝐶 CAR TO KP 𝒉 = 𝒓 × 𝒗 𝑖 = acos ℎ𝑧 ℎ 𝑵 = 𝒌 × 𝒉 ‖𝒌 × 𝒉‖ 𝛺 = acos 𝑁𝑥 𝑁 → 𝛺…