Where does the energy come from that holds an orbit in place?

Question

I found several answers on this site about the question of where gravitational energy comes from, but they are always explained with two objects falling towards each other, something that can be done only once without adding energy to the system again.

I don't see how this applies to a stable orbit. For example in the case of Earth orbiting the sun, where does the energy come from that constantly changes the vector of Earth?

Wikipedia tells me:

Orbits do not decay without some friction-like mechanism which transfers energy from the orbital motion.

But it requires energy to keep Earth in orbit, right? So where does it come from?

Answer 1

in the case of Earth orbitting the sun, where does the energy come from that constantly changes the vector of Earth?

I assume you are referring to the velocity vector $\vec v$ of Earth. The answer is that it does not require energy to turn that vector. An increase in speed $v$ (the magnitude of the vector) requires energy, because that will increase Earth's kinetic energy:

$$K=\frac 12mv^2$$

But turning the velocity vector (changing its direction) without changing the speed, which is what happens in perfectly circular orbits, does not require energy. It requires a force, yes, but not necessarily energy. There is no energy formula like the above which involves the velocity vector.

Where does the energy come from that holds an orbit in place?
But it requires energy to keep Earth in orbit, right? So where does it come from?

Some energy was provided (via fuel in a rocket) to move a satellite up to an orbit, and to give that satellite its necessary initial speed (its initial kinetic energy). Astronomical events billions of years ago provided the energy that formed and positioned all planets, starts and moons and that gave them their initial speeds.

The energy provided in these initial events is energy that will always remain in the orbital system (unless some new forces appear and distort the situation, such as impacts with objects or "friction-like mechanisms" as your Wikipedia quote mentions). By orbital system I mean the system that contains the kinetic and potential energies more or less isolated (such as the Sun-Earth system or the satellite-Earth system).

The forces of gravity from different large bodies such as the sun distort the initial velocity vectors of these moving objects and bodies. They turn them. They may also speed them up or slow them down, in which case those gravitational forces do work. But if they only turn them, then those forces do no work. If the orbits are circular, then all kinetic energy is always with the orbiting body - no energy is taken away or removed; no energy is spent in order to constantly turn them so they stay in their orbit.

Energy is not required to keep up gravitational forces, that's they key point.

Answer 2

To simplify things his answer assumes we are working with the case where the mass of the orbiting object is much less than the mass of the object being orbited around. The ideas here are pretty much the same when this is not the case though.

The work done by a force $\mathbf F$ is given by $$W=\int\mathbf F\cdot\text d\mathbf x$$ i.e. the work depends on the component of the force that is in line with the displacement.

In a circular orbit, the gravitational force is always perpendicular to the displacement, so no work is done or is needed to maintain this orbit. All energies remain constant in this case.

What about elliptical orbits? Well certainly kinetic energy of the orbiting object changes throughout the orbit, but this doesn't mean energy is required to maintain the orbit. As the orbiting object moves father away, gravity does negative work and the object slows down. As the orbiting object moves closer gravity does positive work and the object speeds up. Overall energy is conserved though, as the changes in kinetic energy matches the changes in potential energy (work done by gravity).

There is no need for energy to be put be into the system to maintain the orbit.

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To explicitly comment on some parts of your post:

I found several answers on this site about the question where gravitational energy comes from

Gravitational potential energy is just a different way to describe the work done by gravity. It doesn't really "come from anywhere", as it's actual value is arbitrary. All that really matters is that a change in gravitational energy means work is being done by gravity (positive or negative). But like my answer above, you don't even need to worry about potential energy. That's why I just talked in terms of work done by gravity.

where does the energy come from that constantly changes the vector of Earth?

Note that a change it velocity vector does not mean work is being done. Only a change in its magnitude (speed) does. Like in the elliptical orbit, gravity doing positive or negative work means the speed of the orbiting object changes. But this doesn't mean the energy "comes from somewhere". Gravity does work, the kinetic energy changes, but the total energy of the system remains constant.