Determine the speed v required of an earth satellite at...

Determine the speed $v$ required of an earth satellite at point $A$ for $(a)$ a circular orbit, $(b)$ an elliptical orbit of eccentricity $e=0.1,(c)$ an elliptical orbit of eccentricity $e=0.9,$ and $(d)$ a parabolic orbit. In cases $(b),(c),$ and $(d), A$ is the orbit perigee.

Key Concepts

Parabolic Orbit

A parabolic orbit represents the transition between bound and unbound trajectories. It is a special case where the orbital energy is exactly zero, meaning the object is escaping the gravitational influence of the central body but with zero excess kinetic energy at infinity. The speed calculated for a parabolic trajectory is essentially the escape speed, which depends solely on the gravitational parameter and the distance from the central mass.

Perigee

Perigee refers to the point in an elliptical orbit where the orbiting object is closest to the central body. Because the gravitational pull is strongest at this point, the orbital speed is at its maximum. Understanding perigee is crucial when calculating speeds for elliptical or parabolic orbits, as it sets the conditions under which these computations are made.

Elliptical Orbit

Elliptical orbits are characterized by their eccentricity, a measure of how much the orbit deviates from being circular. In elliptical orbits, the speed of the satellite varies along its path—being highest at perigee (the point closest to the central body) and lowest at apogee. The vis-viva equation can be adapted to determine the orbital speed at any point in an elliptical orbit, taking into account the orbit’s semi-major axis and the current radial distance.

Vis-Viva Equation

The vis-viva equation is a fundamental relation in orbital mechanics that relates the speed of an orbiting body to its distance from the focal point of the orbit and the properties of the orbit. It is applicable to both circular and elliptical orbits, and in the limiting case of a parabolic trajectory, it describes the escape condition where the orbital energy is zero.

Gravitational Parameter

This is a key measure in orbital mechanics defined as the product of the gravitational constant (G) and the mass of the central body (M). It simplifies many orbital equations by encapsulating the effect of gravity into one parameter, frequently denoted as ?. Its value directly affects the dynamics of any orbiting object.

Circular Orbit

A circular orbit is a special case where the distance from the orbiting object to the center of mass remains constant. The orbital speed in a circular orbit is derived by balancing the centripetal force and gravitational force, leading to a constant speed that depends only on the gravitational parameter and the orbital radius.

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