KAHIBARO
Table of Contents
Energy of an Orbiting Body
When an object moves in orbit around a planet, moon, or star, its energy is a combination of kinetic energy and gravitational potential energy. Orbital energy tells us how tightly the object is bound to the central body and helps explain why satellites stay in orbit, why some orbits are circular and others are elliptical, and how much energy is needed to move from one orbit to another.
For a body of mass $m$ orbiting a much larger body of mass $M$, the total mechanical energy is
$$
E = K + U
$$
where $K$ is kinetic energy and $U$ is gravitational potential energy.
The gravitational potential energy at a distance $r$ from the center of the larger body is
$$
U = -\frac{GMm}{r}
$$
The negative sign is very important. It means the object is gravitationally bound. Zero potential energy is defined when the two bodies are infinitely far apart.
For gravity, the potential energy is
$$
U = -\frac{GMm}{r}
$$
not $+mgh$, except near Earth's surface as an approximation.
Circular Orbit Energy
For a circular orbit, the gravitational force provides the centripetal force. From this relation, the orbital speed is
$$
v = \sqrt{\frac{GM}{r}}
$$
The kinetic energy is then
$$
K = \frac{1}{2}mv^2 = \frac{1}{2}m\frac{GM}{r} = \frac{GMm}{2r}
$$
The potential energy is
$$
U = -\frac{GMm}{r}
$$
So the total orbital energy becomes
$$
E = K + U = \frac{GMm}{2r} - \frac{GMm}{r} = -\frac{GMm}{2r}
$$
For a circular orbit,
$$
K = \frac{GMm}{2r}, \qquad U = -\frac{GMm}{r}, \qquad E = -\frac{GMm}{2r}
$$
The total energy is negative for a bound orbit.
This result shows that the total energy depends only on the orbital radius. A smaller radius means more negative energy, so the satellite is more tightly bound.
Meaning of Negative Orbital Energy
A negative total energy means the object does not have enough energy to escape the gravitational attraction of the central body. Energy must be added to the object to bring its total energy up to zero.
If
$$
E < 0
$$
the orbit is bound.
If
$$
E = 0
$$
the object is just at the escape condition.
If
$$
E > 0
$$
the object is unbound and can move away forever.
This is why satellites in normal orbits have negative total energy.
Elliptical Orbit Energy
Not all orbits are circular. Many are elliptical. In that case, the speed changes from point to point, and both kinetic and potential energy change during the motion. However, for an ideal orbit with no drag and no thrust, the total mechanical energy stays constant.
For an elliptical orbit, the total energy is
$$
E = -\frac{GMm}{2a}
$$
where $a$ is the semi-major axis of the ellipse.
This formula looks very similar to the circular orbit formula. In fact, a circular orbit is just a special case of an ellipse with constant radius $r = a$.
For any bound Kepler orbit,
$$
E = -\frac{GMm}{2a}
$$
where $a$ is the semi-major axis.
This means that the total orbital energy does not depend directly on the current position in the orbit, but on the size of the orbit.
Specific Orbital Energy
It is often useful to divide the total orbital energy by the satellite mass $m$. This gives the specific orbital energy, which is energy per unit mass:
$$
\varepsilon = \frac{E}{m}
$$
For a circular orbit,
$$
\varepsilon = -\frac{GM}{2r}
$$
For an elliptical orbit,
$$
\varepsilon = -\frac{GM}{2a}
$$
Using specific energy makes calculations easier because the satellite mass cancels out.
Comparison of Energy Terms
The relation between kinetic, potential, and total energy in a circular orbit is especially simple.
| Quantity | Formula | Sign |
|---|---|---|
| Kinetic energy | $K = \frac{GMm}{2r}$ | Positive |
| Potential energy | $U = -\frac{GMm}{r}$ | Negative |
| Total energy | $E = -\frac{GMm}{2r}$ | Negative |
From these formulas, we can also see that
$$
U = -2K
$$
and therefore
$$
E = -K = \frac{U}{2}
$$
In a circular orbit,
$$
U = -2K, \qquad E = -K = \frac{U}{2}
$$
These relations are very useful for quick calculations.
Energy Needed to Change Orbits
To move a satellite to a higher orbit, energy must be added. A higher orbit has a larger radius or semi-major axis, so its total energy is less negative.
For example, if a satellite moves from radius $r_1$ to radius $r_2$ in circular orbits, the change in total energy is
$$
\Delta E = E_2 - E_1 = -\frac{GMm}{2r_2} + \frac{GMm}{2r_1}
$$
If $r_2 > r_1$, then $\Delta E > 0$, so energy must be supplied.
If a satellite moves to a lower orbit, the total energy becomes more negative, which means energy must be removed.
This sometimes feels strange because satellites in lower orbit move faster, but they still have lower total energy because the decrease in potential energy is larger than the increase in kinetic energy.
Visual Idea of Orbital Energy
A useful way to think about orbital energy is to imagine a deep gravitational well. Close to the planet, the object sits deeper in the well and has more negative energy. Far away, the energy rises toward zero.
The curve shows the gravitational potential energy. Circular orbit energy lies below zero, showing that the satellite is bound.
Example
Suppose a satellite of mass $m$ orbits Earth in a circular orbit of radius $r$ from Earth's center. Its total energy is
$$
E = -\frac{GM_{\oplus}m}{2r}
$$
If the orbital radius is doubled to $2r$, the new energy is
$$
E' = -\frac{GM_{\oplus}m}{2(2r)} = -\frac{GM_{\oplus}m}{4r}
$$
So the new orbit has a higher energy because it is less negative.
The change in energy is
$$
\Delta E = E' - E = -\frac{GM_{\oplus}m}{4r} + \frac{GM_{\oplus}m}{2r}
= \frac{GM_{\oplus}m}{4r}
$$
This positive result means energy must be added to raise the orbit.
Key Physical Insight
Orbital energy is not just about how fast a satellite moves. It is about the balance between motion and gravitational binding. A satellite in orbit constantly exchanges kinetic and potential energy if the orbit is elliptical, but the total remains fixed unless an external force acts.
The central idea is that bound orbits have negative total energy, and the size of the orbit determines how negative that energy is.
Main formulas for orbital energy:
$$
U = -\frac{GMm}{r}
$$
$$
E_{\text{circular}} = -\frac{GMm}{2r}
$$
$$
E_{\text{elliptical}} = -\frac{GMm}{2a}
$$
A more negative energy means a more tightly bound orbit.