A satellite in orbit has kinetic energy because it is moving and gravitational potential energy because it is in the gravitational field of a central body.
The total mechanical energy is
\[
E=K+U.
\]
For a bound satellite orbit, this total energy is negative.
1. Gravitational potential energy
With the zero of gravitational potential energy chosen at infinity,
\[
\boxed{
U=-\frac{GMm}{r}
}.
\]
Here:
- \(G\) is the gravitational constant,
- \(M\) is the central mass,
- \(m\) is the satellite mass,
- \(r\) is the distance from the center of the central body.
The negative sign means the satellite is gravitationally bound at finite distance.
2. Circular orbit energy
For a circular orbit, gravity supplies centripetal force:
\[
\frac{GMm}{r^2}=\frac{mv^2}{r}.
\]
Canceling \(m\) and simplifying gives
\[
v^2=\frac{GM}{r}.
\]
The kinetic energy is
\[
K=\frac12mv^2.
\]
Substitute \(v^2=GM/r\):
\[
K=\frac12m\left(\frac{GM}{r}\right)
=
\boxed{
\frac{GMm}{2r}
}.
\]
The total mechanical energy is
\[
E=K+U.
\]
\[
E=
\frac{GMm}{2r}
-
\frac{GMm}{r}.
\]
\[
\boxed{
E_{\mathrm{circ}}=-\frac{GMm}{2r}
}.
\]
3. Binding energy
The binding energy of a satellite in a circular orbit is the positive amount of energy that must be added to reach the escape threshold \(E=0\).
Since a bound circular orbit has
\[
E_{\mathrm{circ}}=-\frac{GMm}{2r},
\]
the binding energy is
\[
\boxed{
B=-E_{\mathrm{circ}}=\frac{GMm}{2r}
}.
\]
A satellite in a lower orbit has a larger binding energy because it is deeper in the gravitational potential well.
4. Energy required to move to a higher circular orbit
Let the initial circular orbit radius be \(r_1\), and the target circular orbit radius be \(r_2\).
The initial and final total energies are
\[
E_1=-\frac{GMm}{2r_1},
\qquad
E_2=-\frac{GMm}{2r_2}.
\]
The change in orbital mechanical energy is
\[
\Delta E=E_2-E_1.
\]
Substitute the circular-orbit energies:
\[
\Delta E
=
-\frac{GMm}{2r_2}
+
\frac{GMm}{2r_1}.
\]
Therefore,
\[
\boxed{
\Delta E=
\frac{GMm}{2}
\left(
\frac{1}{r_1}
-
\frac{1}{r_2}
\right)
}.
\]
If \(r_2>r_1\), then \(\Delta E>0\), so energy must be added.
If \(r_2
5. Energy required to escape
Escape occurs at the threshold
\[
E=0.
\]
If the satellite starts in a circular orbit with total energy
\[
E_1=-\frac{GMm}{2r_1},
\]
then the energy that must be added to reach escape is
\[
\Delta E_{\mathrm{escape}}
=
0-E_1.
\]
\[
\boxed{
\Delta E_{\mathrm{escape}}
=
\frac{GMm}{2r_1}
}.
\]
This is exactly the binding energy of the initial circular orbit.
6. Escape speed
Escape speed is found by setting the total energy to zero:
\[
0=\frac12mv_{\mathrm{esc}}^2-\frac{GMm}{r}.
\]
Solving for \(v_{\mathrm{esc}}\) gives
\[
\boxed{
v_{\mathrm{esc}}=\sqrt{\frac{2GM}{r}}
}.
\]
Circular speed is
\[
v_{\mathrm{circ}}=\sqrt{\frac{GM}{r}}.
\]
Therefore,
\[
\boxed{
v_{\mathrm{esc}}=\sqrt{2}\,v_{\mathrm{circ}}
}.
\]
7. Hohmann transfer preview
A Hohmann transfer is an efficient two-burn transfer between two coplanar circular orbits.
The transfer path is half of an ellipse tangent to both circular orbits.
If the circular orbit radii are \(r_1\) and \(r_2\), the transfer ellipse semi-major axis is
\[
\boxed{
a_t=\frac{r_1+r_2}{2}
}.
\]
The total mechanical energy while on the transfer ellipse is
\[
\boxed{
E_t=-\frac{GMm}{2a_t}
}.
\]
The circular speeds are
\[
v_1=\sqrt{\frac{GM}{r_1}},
\qquad
v_2=\sqrt{\frac{GM}{r_2}}.
\]
The transfer speeds at the two burn points come from the vis-viva equation:
\[
v_t=\sqrt{
GM\left(\frac{2}{r}-\frac{1}{a_t}\right)
}.
\]
Therefore,
\[
v_{t1}=
\sqrt{
GM\left(\frac{2}{r_1}-\frac{1}{a_t}\right)
},
\]
\[
v_{t2}=
\sqrt{
GM\left(\frac{2}{r_2}-\frac{1}{a_t}\right)
}.
\]
The two ideal impulse changes are
\[
\Delta v_1=v_{t1}-v_1,
\qquad
\Delta v_2=v_2-v_{t2}.
\]
The total Hohmann transfer speed change is
\[
\boxed{
|\Delta v|_{\mathrm{total}}
=
|\Delta v_1|+|\Delta v_2|
}.
\]
8. Hohmann transfer time
The Hohmann transfer uses half of the transfer ellipse, so the transfer time is half the orbital period of the transfer ellipse:
\[
t_t=\frac12
\left(
2\pi\sqrt{\frac{a_t^3}{GM}}
\right).
\]
Therefore,
\[
\boxed{
t_t=\pi\sqrt{\frac{a_t^3}{GM}}
}.
\]
9. Worked example: LEO to GEO transfer
Suppose a \(1000\ \mathrm{kg}\) satellite moves from a low Earth orbit at \(400\ \mathrm{km}\) altitude to a geostationary orbit at about \(35{,}786\ \mathrm{km}\) altitude.
Use
\[
M_{\oplus}=5.9722\times10^{24}\ \mathrm{kg},
\qquad
R_{\oplus}=6.371\times10^6\ \mathrm{m}.
\]
The orbital radii are approximately
\[
r_1=R_{\oplus}+400\ \mathrm{km}
\approx6.771\times10^6\ \mathrm{m},
\]
\[
r_2=R_{\oplus}+35786\ \mathrm{km}
\approx4.216\times10^7\ \mathrm{m}.
\]
The initial circular-orbit energy is
\[
E_1=-\frac{GMm}{2r_1}.
\]
The final circular-orbit energy is
\[
E_2=-\frac{GMm}{2r_2}.
\]
The energy required to raise the circular orbit is
\[
\Delta E
=
\frac{GMm}{2}
\left(
\frac{1}{r_1}
-
\frac{1}{r_2}
\right).
\]
Since \(r_2>r_1\), this value is positive. The target GEO orbit is less tightly bound, so the satellite must gain mechanical orbital energy.
10. Energy level interpretation
| Energy level |
Formula |
Interpretation |
| Lower circular orbit |
\(E_1=-GMm/(2r_1)\) |
More negative, more tightly bound |
| Higher circular orbit |
\(E_2=-GMm/(2r_2)\) |
Less negative, less tightly bound |
| Transfer ellipse |
\(E_t=-GMm/(2a_t)\) |
Intermediate energy level during Hohmann transfer |
| Escape threshold |
\(E=0\) |
Boundary between bound and unbound motion |
11. Summary formulas
| Goal |
Formula |
Use |
| Circular total energy |
\(E=-GMm/(2r)\) |
Energy of a satellite in a circular orbit |
| Binding energy |
\(B=GMm/(2r)\) |
Energy needed to escape from a circular orbit |
| Orbit raising energy |
\(\Delta E=GMm(1/r_1-1/r_2)/2\) |
Net mechanical energy change between circular orbits |
| Transfer semi-major axis |
\(a_t=(r_1+r_2)/2\) |
Size of Hohmann transfer ellipse |
| Transfer energy |
\(E_t=-GMm/(2a_t)\) |
Mechanical energy on the transfer ellipse |
| Transfer speed |
\(v_t=\sqrt{GM(2/r-1/a_t)}\) |
Speed on transfer ellipse at a burn point |
| Transfer time |
\(t_t=\pi\sqrt{a_t^3/(GM)}\) |
Time for half the transfer ellipse |
12. Assumptions and limits
- The central body is treated as spherical or point-like.
- Only Newtonian gravity is included.
- Atmospheric drag, thrust arcs, finite burn time, and perturbations are ignored.
- The Hohmann transfer assumes two coplanar circular orbits.
- The energy change \(\Delta E\) is mechanical orbital energy change, not the full chemical energy required by a real rocket.
- Real mission design must include engine efficiency, propellant mass, inclination changes, launch constraints, and station-keeping.
Key idea: a satellite in a higher circular orbit has less negative total energy, so raising an orbit requires adding mechanical energy even though the final circular speed may be lower.
