By Tolson R.H.
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Extra resources for Space Flight Mechanics a.k.a. Astrodynamics
Sample text
Develop the equivalent of equations (3-32) for parabolic orbits. 3-4. Develop the equivalent of equations (3-32) for hyperbolic orbits. 3-5. Verify the Φ(2,2) term in equation (3-33) using spherical trigonometry relations. 14 Astronautics Toolbox 1. Write a procedure that returns the rotation matrix (3 by 3) for an arbitrary [3,1,3] set of rotations [α,β,γ], Φ=Rotate313(α,β,γ,ichk). 2. Write a procedure to solve Barker's equation (3-16), f=Barker(t,τ,p,µ,ichk). Assume t is (n by 1). 3. Write a procedure to solve Kepler's equation (3-14) for elliptic motion using Newton-Raphson iteration, E=Kepler(M,e,tol,ichk).
Table 2-1. 4). The argument of perihelion is ω and λ o = M o + ϖ where Mo is the mean anomaly at J2000. The four angles are in degrees. Note that the ephemeris is for the Earth-Moon barycenter. 327124 x 1011 km3/s2. 6 Problems 2-1. Show that the gravitational force on a particle inside a homogeneous spherical shell vanishes. 2-2. 7 Astronautics Toolbox 1. Write a function OE=PlanetOE(JD,PlanetNum), that returns the six orbital elements(OE) at the Julian date=JD. 9 to identify the planet and include the centennial rates.
During the Apollo era, the problem received renewed interest since the two dominate forces action on the spacecraft were due to the Earth and the Moon. Even for this system, requiring 18 integrals, a closed-form solution of the general problem does not appear feasible and there are no known integrals beyond those discussed in Chapter 2. e. the characteristics of the motion are very sensitive to the initial conditions. There exist, however, particular solutions of the three-body problem obtained by Lagrange in 1772, which will be discussed later.