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Kinetic energy is the energy an object possesses due to its motion. In the context of celestial mechanics, when an object is in orbit, its kinetic energy keeps it moving along its path around the central body, counterbalancing the gravitational pull. For a satellite or rocket in a circular orbit, this energy is calculated using the formula

\[\begin{equation} KE = \frac{1}{2} m v^2 \end{equation}\]

where \(m\) represents its mass and \(v\) its orbital velocity. Crucially, the speed must be high enough to sustain the orbit but not too high to escape the central body's gravitational pull.

For a circular orbit, where gravity provides the centripetal force, the velocity can be expressed through gravitational parameters as

\[\begin{equation} v^2 = \frac{GM}{r} \end{equation}\]

with \(G\) as the gravitational constant, and \(M\) as the mass of the central body, such as Earth. Here, \(r\) stands for the orbit's radius. From the provided exercise, the kinetic energy changes as the radius of the orbit changes, demonstrating its inverse relationship with the orbit's radius: as the radius increases, the kinetic energy decreases, since the velocity is lower at a higher orbit.

Gravitational potential energy in the context of orbital mechanics is the energy stored due to an object's position in a gravitational field. It is given by

\[\begin{equation} U = - \frac{GMm}{r} \end{equation}\]

The negative sign indicates that work needs to be done against the gravitational pull to move objects such as a rocket to larger orbits, hence increasing its potential energy.

The interplay between gravitational potential energy and kinetic energy is crucial for understanding orbits. While the kinetic energy resists the gravitational pull by keeping the object in motion, the gravitational potential energy pulls the object toward the central body due to gravity.

In the scenario of the exercise, when the rocket's orbit radius is increased, the gravitational potential energy increases because the rocket is further away from Earth and has 'climbed out' of the gravitational well to a certain extent. This increase in potential energy reflects the work required to reach a higher orbit.

The work done by rocket engines, or any other force in altering an orbit, can be thought of as the energy required to transition an object from one orbit to another. This work is equivalent to the change in the object's total mechanical energy, which is the sum of its kinetic and potential energy.

Following the analysis of the exercise, the formula to express work done is

\[\begin{equation} W = \Delta KE + \Delta U \end{equation}\]

Work thus is done when the rocket engines fire and change the rocket's orbit. It's important to note that the total mechanical energy will be conserved if there are no external forces doing work on the system; however, in this case, the rocket engine does external work.

In the situation of the example, the kinetic energy decreases but the gravitational potential energy increases. As a result, the work done is effectively the energy invested into the system to increase the potential energy, and it can either come from the chemical energy of fuel in the case of rocket engines or from other sources like gravity assists in space maneuvers.