Orbital Mechanics
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The energy and angular momentum of a particle inside a central potential are given by:
E = ½*μ *(dr/dt)^2 + V(r) L = μ* r^2 *(dθ/dt)
V(r) = - (G*μ*M)/r + L^2/(2*μ*r^2) - (G*L^2*M)/(r^3*μ*c^2)
μ is the reduced mass and c is the speed of light. (Use units where c=1).
a) Solve these two equations for dr/dt and dθ/dt and show that:
dr/dt = +/- [( 2/μ)*(E - V(r))]^1/2 dθ/dt = L/(μ*r^2)
b) Write the equation of a) in a μ independent form by introducing e = E/μ and l = L/μ.
c) Use units such that G = -M = c = 1, and let e = -.02, l = 4.5. Let the initial position be given by r(t=0) = a, θ(t=0) = 0, a = -GM/(2e). Compute the initial velocities dr/dt(t=0) and dθ/dt(t=0) under the assumption both are positive.
d) Plot V(r)
Note: a = -GM/(2e)
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Step by step solution with explanation.
Education
- BSc , Wuhan Univ. China
- MA, Shandong Univ.
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