Interactive physics simulator
Satellite Motion
Understand satellite dynamics in circular and elliptical orbits. Observe apogee and perigee variations, and trace kinetic vs potential energy conservation.
Satellite Motion Simulator
Launch satellites into Earth orbit, analyze apogee/perigee telemetry, and observe kinetic and potential energy transfers.
Live Telemetry
- Current Altitude
- 600 km
- Current Speed
- 7.56 km/s
- Perigee Altitude
- 600 km
- Apogee Altitude
- 600 km
- Kinetic Energy (KE)
- 28.6 MJ/kg
- Potential Energy (PE)
- -57.2 MJ/kg
- Total Mechanical (E)
- -28.6 MJ/kg
- Orbital Period
- 96.7 min
Satellite Orbits and Mechanical Energy
A satellite moves in an orbit determined by its initial launch energy. In elliptical orbits, altitude and speed vary continuously: the satellite slows down as it climbs away from the planet, transferring kinetic energy to potential energy, and speeds up as it descends back. Total mechanical energy, E = KE + PE, is conserved.
Orbital Geometry
Satellite paths are conic sections:
- Circular Orbit: Speed is constant and equal to vorb.
- Elliptical Orbit: Speed varies between perigee (fastest) and apogee (slowest).
- Parabolic Orbit: Speed equals vesc; the satellite escapes gravity on an open path.
- Hyperbolic Orbit: Speed exceeds vesc.
Conservation of Energy
For circular orbits, kinetic energy is exactly half the magnitude of potential energy: KE = -½PE. Total energy equals half the potential energy: E = ½PE = -GMm/2r.
Apogee vs Perigee
| Property | Perigee (Closest) | Apogee (Farthest) |
|---|---|---|
| Altitude (r) | Minimum | Maximum |
| Orbital Speed (v) | Maximum | Minimum |
| Kinetic Energy (KE) | Maximum | Minimum |
| Potential Energy (PE) | Minimum (most negative) | Maximum (least negative) |
| Total Energy (E) | Conserved (Constant) | Conserved (Constant) |
Solved Examples
A satellite of mass m = 200 kg is in a circular orbit around Earth at an altitude of 600 km. Find its potential energy, kinetic energy, and total mechanical energy. (Earth mass M = 5.972 × 1024 kg, radius R = 6,371 km).
- Find orbital radius: r = 6,371 km + 600 km = 6,971 km = 6.971 × 106 m.
- Calculate Potential Energy (PE): PE = -GMm/r. G = 6.6743 × 10-11 N·m²/kg².
- PE = -((6.6743 × 10-11 × 5.972 × 1024 × 200) / (6.971 × 106)) ≈ -1.144 × 1010 Joules = -11.44 GJ.
- Find circular orbital velocity: vorb = √(GM/r) ≈ 7.562 km/s = 7,562 m/s.
- Calculate Kinetic Energy (KE): KE = 0.5 × m × v² = 0.5 × 200 × (7,562)² ≈ 5.718 × 109 Joules = 5.72 GJ. (Note: in a circular orbit, KE = -0.5 × PE).
- Calculate Total Energy (E): E = KE + PE = 5.72 GJ - 11.44 GJ = -5.72 GJ.
Answer: PE ≈ -11.44 GJ, KE ≈ 5.72 GJ, Total Energy E ≈ -5.72 GJ
A satellite in an elliptical orbit has a perigee speed of 9.50 km/s at an altitude of 300 km. If its speed at apogee is 6.20 km/s, calculate the altitude of the apogee. (Earth radius R = 6,371 km).
- Use Kepler’s Second Law (conservation of angular momentum): rperigee × vperigee = rapogee × vapogee.
- Find perigee radius: rperigee = 6,371 km + 300 km = 6,671 km.
- Substitute values: 6,671 km × 9.50 km/s = rapogee × 6.20 km/s.
- Solve for apogee radius: rapogee = (6,671 × 9.50) / 6.20 ≈ 10,221.7 km.
- Find apogee altitude above surface: altitude = rapogee - R = 10,221.7 km - 6,371 km ≈ 3,851 km.
Answer: Apogee Altitude ≈ 3,851 km
Show that the total specific energy of a satellite orbiting in a circular orbit is negative, and explain what this physical sign represents.
- Specific kinetic energy (per unit mass): ke = 0.5 × v² = 0.5 × (GM/r) = GM/(2r).
- Specific potential energy (per unit mass): pe = -GM/r.
- Specific total energy: e = ke + pe = GM/(2r) - GM/r = -GM/(2r).
- Because G, M, and r are positive, the total specific energy e is always negative for bound orbits.
- The negative sign represents that the system is gravitationally bound, meaning the satellite does not have enough energy to escape to infinity on its own.
Answer: e = -GM/(2r) (bound orbit)
Common Pitfalls
- Thinking total mechanical energy fluctuates during elliptical orbits.
- Forgetting that potential energy is a negative quantity in gravitational systems: PE = -GMm/r.
- Believing satellites require constant thrust to stay in orbit (inertia and gravity maintain motion).
Kepler's Second Law
A satellite sweeps out equal areas in equal times, which mathematically means orbital angular momentum is conserved:
This is why satellites speed up at perigee when r is smaller and slow down at apogee when r is larger.
Practice Questions
1. Why do satellites in elliptical orbits move fastest at perigee?
As a satellite falls closer to the planet (toward perigee), its potential energy decreases because it descends into a deeper gravity well. By conservation of mechanical energy, this lost PE is converted into kinetic energy, causing its speed to increase.
2. What does a total mechanical energy E = 0 represent for a launched satellite?
A total energy of exactly zero represents a parabolic escape trajectory. The satellite has just enough kinetic energy to escape to infinity, where it will come to rest.
3. Why does a geostationary satellite have to orbit exactly above the equator?
To remain fixed over one spot, its orbital plane must match the Earth’s rotational plane (which is perpendicular to the axis of rotation). If it were inclined, it would move north and south in the sky during its orbit.
4. How is angular momentum conserved in satellite motion?
Since gravity is a central force pointing directly toward the planet center, it exerts zero torque on the satellite. Thus, orbital angular momentum L = m × r × v × sin(θ) remains perfectly constant throughout the orbit.
FAQ
Frequently Asked Questions
What is satellite motion?
Satellite motion refers to the movement of a smaller body (natural moon or artificial satellite) orbiting around a much more massive central body (like a planet or star) under the influence of gravity.
What are Low Earth Orbit (LEO) and Geostationary Earth Orbit (GEO)?
LEO is a low altitude orbit (160 to 2,000 km above Earth) with speeds of ~7.8 km/s and periods of ~90 minutes. GEO is a high altitude orbit at 35,786 km above the equator where the satellite's orbital period matches Earth's rotation (24 hours), keeping it fixed over one spot at a speed of ~3.07 km/s.
How is energy conserved in satellite orbits?
The total mechanical energy of an orbiting satellite (E = KE + PE) is conserved. Kinetic energy is KE = 0.5 * m * v², and gravitational potential energy is PE = -G * M * m / r. In circular orbits, both KE and PE are constant. In elliptical orbits, they swap back and forth (KE increases and PE decreases at perigee, and vice versa at apogee) but their sum remains constant.
What are apogee and perigee in satellite motion?
Perigee is the point in an orbit that is closest to the central body, where the gravitational force is strongest and the satellite moves fastest. Apogee is the point that is farthest from the central body, where the gravitational force is weakest and the satellite moves slowest.
What is a Geostationary Transfer Orbit (GTO)?
GTO is a highly elliptical orbit used to transition a satellite from a low altitude parking orbit to a high geostationary orbit. It has a low perigee (~250 km) and a high apogee (~35,786 km).
What is a Molniya orbit?
A Molniya orbit is a highly eccentric, highly inclined elliptical orbit. Satellites spend most of their time (nearly 8 hours of a 12-hour period) slowly moving near apogee over high latitudes, providing communications coverage to polar regions.
How does a satellite change its orbit?
A satellite changes its orbit by firing its thrusters. Firing forward (prograde) at perigee increases kinetic energy, raising the altitude of the apogee on the opposite side. Firing backward (retrograde) lowers the opposite side's altitude.
What force keeps a satellite in orbit?
The gravitational force between the planet and the satellite acts as the centripetal force, continuously pulling the satellite toward the center of the planet and bending its path into a curve.
Why don't satellites fall to Earth?
Satellites do not fall to Earth because their forward (tangential) velocity is so high that the curve of their fall matches the curve of the Earth. They are constantly falling 'around' the Earth rather than 'into' it.
Does atmospheric drag affect satellite motion?
Yes. Satellites in LEO experience a tiny amount of atmospheric drag from thin air, which slowly drains their energy and causes orbital decay, eventually leading to reentry and burning up unless they periodically boost their altitude.
What is Kepler's Second Law in satellite motion?
Kepler's Second Law states that a line connecting a satellite to the planet sweeps out equal areas in equal time intervals. This explains why satellites speed up when they are closer (perigee) and slow down when they are farther away (apogee).
How does the satellite energy chart in the simulator work?
The simulator tracks kinetic energy and potential energy at every point along the orbit. The live stacked bar graph shows that as PE goes up, KE goes down, and vice versa, keeping the total height representing total energy perfectly constant.