Orbit of a Launched Satellite

Kepler observed that the path of a satellite in orbit around a heavy mass traces out an ellipse. More generally, an orbit in the absence of a third body is always a conic section (i.e., an ellipse, parabola, or hyperbola). For a parabolic orbit, the eccentricity equals 1 for any value of the initial launch speed. This speed along with the planet's mass, the initial altitude, and the gravitational constant, , determine the eccentricity of the conic section traced by the satellite's orbit. If the orbit is closed, then the speed is always greatest closest to the planet and smallest at the maximum distance from the planet. This is a consequence of the conservation of angular momentum.


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For a circular or elliptical orbit, the potential energy is twice the negative of the kinetic energy, as a result of the virial theorem. The eccentricity is zero for a circular orbit and for an elliptical orbit. For a hyperbolic orbit, . In this Demonstration, the largest eccentricity is about 21. As the eccentricity approaches 1, the initial launch speed should approach zero. This is why small trajectories at the Earth's surface can be approximated by parabolas. When , the shape of the orbit degenerates into a parabola, moving at the escape velocity.
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