Orbital Mechanics
Satellite motion, circular V = √(GM/R) V = √(gR) T = 2π√[R³/GM] g = GM/R² T is period of satellite in sec V = velocity in m/s g = acceleration of gravity in m/s² (9.8 m/s² at ground level) G = 6.673e-11 Nm²/kg² M is mass of central body in kg R is radius of orbit in m elliptic orbit T = 2π√[a³/GM] T is period of satellite in sec a = is length of orbit's semi-major axis (half the major axis) G = 6.673e-11 Nm²/kg² M is mass of central body in kg Two objects orbiting each other circular orbits T = 2π√[r³/G(M₁+M₂)] r is distance between them Two objects orbiting each other P = 2π√[a³/G(M₁+M₂)] P is orbital period in seconds M₁ M₂ are the masses of the two bodies G = 6.673e-11 Nm²/kg² a is the sum of the semi-major axes of the ellipses in which the centers of the bodies move, or equivalently, the semi-major axis of the ellipse in which one body moves, in the frame of reference with the other body at the origin. earth radius 6,371 km = 6.37e6 meters earth mass 5.97e24 kg earth GM = 3.98e14 earth max speed for orbit 7900 m/s moon radius 1,737,000 m moon mass 7.35e22 kg sun mass 1.9891e30 kg distance earth to sun 1.5e11 m Sun surface gravity 274 m/s² Orbital energy potential energy plus KE of orbit is h is altitude above surface m is mass of Satellite M is mass of planet G = 6.673e-11 Nm²/kg² E = –GmM / (2(R+h)) or if h is distance from center of earth E = –GmM / (2h) (energy required for launch is negative of this) Escape speed V₀ = √(2GM/r) V₀ = √(2gr) (alternate) G = 6.673e-11 Nm²/kg² where M is mass of body and r is distance from center of body g is gravity at the surface Relativistic escape velocity V₀ = √[ (2GM/r) – (GM/rc)² ] Earth 11.2 km/s Earth+Sun 42.1 km/s Venus 10.3 km/s Venus+Sun 49.5 km/s Mars 5.0 km/s Mars+Sun 34.1 km/s Moon 2.4 km/s Moon+Earth 1.4 km/s speed needed to reach a certain height h V = √(2GM(1/Re - 1/(Re+h))) Change in energy when a satellite changes altitude How much work must be done to move the satellite into another circular orbit that is higher above the surface of the Earth? Satellite change in energy with height Assuming the satellite is to be boosted to a new height r₂ Gravitational potential energy (to center of earth) new orbit(2) has a higher PE than old one(1), so change is positive PE = G m₁m₂/r earth GM = 3.98e14 ΔPE = (GMm)(1/r₁ – 1/r₂) KE also changes. Get velocity at each height. New orbit(2) has lower speed, so change is negative v = √(GM/R) V₁ = √(GM/r₁) V₂ = √(GM/r₂) ΔKE = –½m(V₁² – V₂²) ΔKE = –½mGM(1/r₁ – 1/r₂) adding the two ΔE = (GM)(m)(1/r₁ – 1/r₂)– ½mGM(1/r₁ – 1/r₂) ΔE = ½mGM(1/r₁ – 1/r₂) |
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