# work of body remaining still in gravitational field

How to quantify power output of a statite?

It's a kind of (non-)satellite, that remains immobile in relation to the central body (e.g. hovering over one of the poles of a planet) through continuous counteracting the gravitational pull through its propulsion. (the linked article suggests solar sail, but other means have been proposed; ion propulsion based Sun statite could maintain position for months.)

If we take work as force over distance, since statite remains immobile, work understood that way is zero. This approach is impractical here.

Yet indubitably fuel is burnt, propellant accelerated, a certain energy output over time is required to maintain status quo.

How should I calculate this work, energy and power, knowing the local gravitational acceleration and mass of the statite?

(let's assume the statite mass change over time of the experiment is small to avoid rocket equation problems).

While the statite remains stationary, it has to eject mass (like a rocket motor) in order to maintain altitude. The work done is computed by considering the momentum exchange needed per unit time. If the force of gravity you are counteracting is $F$, then the momentum of matter accelerated per unit time is also $F$ (since $\Delta P = F\Delta t$, it follows that $F=\frac{\Delta P}{\Delta t}$)
Now the energy contained in the ejected mass is $\frac12 m v^2 = \frac{P^2}{2m}$. It follows that the larger the ejected mass, the lower the energy needed. This is why the Sikorsky challenge was won by a human-powered helicopter with a giant span - it could move a very large amount of air (mass) very slowly (low energy needed).