Abstract
Transition from elliptic to hyperbolic orbits in the two-body problem with slowly decreasing mass is investigated by means of asymptotic approximations.
Analytical results by Verhulst and Eckhaus are extended to construct approximate solutions for the true anomaly and the eccentricity of the osculating orbit if the initial conditions are nearly-parabolic. It becomes clear that the eccentricity will monotonously increase with time for all mass functions satisfying a Jeans-Eddington relation and even for a larger set of functions. To illustrate these results quantitatively we calculate the eccentricity as a function of time for Jeans-Eddington functionsn=0(1) 5 and 18 nearly-parabolic initial conditions to find that 93 out of 108 elliptic orbits become hyperbolic.
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References
Boersma, J.: 1961,Bull. Astron. Inst. Neth. 15, 291–301.
Verhulst, F.: 1969,Bull. Astron. Inst. Neth. 20, 215–21.
Verhulst, F.: 1972,Celes. Mech. 5, 27–36.
Verhulst, F. and Eckhaus, W.: 1970,Intern. J. Nonlinear Mech. 5, 617–24.
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Van Der Laan, L., Verhulst, F. The transition from elliptic to hyperbolic orbits in the two-body problem by slow loss of mass. Celestial Mechanics 6, 343–351 (1972). https://doi.org/10.1007/BF01231477
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DOI: https://doi.org/10.1007/BF01231477