Final answer:
The transition from an elliptical orbit to a circular orbit involves applying Kepler's Laws of Planetary Motion, which require a velocity change at specific points in the orbit to maintain a constant radius throughout the new circular orbit.
Step-by-step explanation:
The question pertains to the transition of an object from an initial elliptical orbit with a periapsis (rp) at a distance of 'a' and apoapsis (ra) at a distance of '3a', to a final circular orbit. To understand this transition, one must apply Kepler's Laws of Planetary Motion. Kepler's First Law indicates that the path of a planet or an object around the Sun is an ellipse, with the Sun at one of the two foci. Also, a circle is a special case of an ellipse where the two foci coincide, making all points on the orbit equidistant from the center.
Kepler's Second Law asserts that the line connecting a planet to the Sun sweeps out equal areas in equal times. This implies that the planet's speed varies along its orbit, moving faster near the periapsis and slower near the apoapsis. When migrating to a circular orbit from an elliptical orbit, the object will need to increase its velocity at the apoapsis or decrease it at the periapsis, ensuring that the radius remains constant throughout the orbit, which means changing the orbital shape.
Using Kepler's Third Law, which states that for any planet orbiting the Sun, the square of the period of the orbit is directly proportional to the cube of the semi-major axis of its orbit, we can deduce that achieving a circular orbit could theoretically require a change in energy and hence a change in velocity at certain points along the initial orbit.