Eccentricity and Speed: Kepler's second law, visualized
Delta-V Academy / Learn / Lesson 3
Same orbit, different speeds. Drag eccentricity, watch the satellite race through perigee and crawl through apogee.
In a circular orbit, the satellite moves at a constant speed. The moment you stretch the orbit into an ellipse, that stops being true. The satellite speeds up as it falls toward Earth (perigee) and slows down as it climbs away (apogee). This is Kepler's second law: the line from Earth to the satellite sweeps out equal areas in equal times. The more eccentric the orbit, the more dramatic the speed difference. A Molniya satellite moves 10 km/s at perigee and 1.5 km/s at apogee — same orbit, seven times the speed.
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What you'll learn
- Kepler's second law: equal areas in equal times
- The vis-viva equation: v² = μ × (2/r − 1/a)
- Why Molniya satellites dwell over Russia for 8 hours per 12-hour orbit
- How eccentricity changes velocity at perigee vs apogee
- Why highly elliptical orbits (HEO) are useful for high-latitude coverage
The vis-viva equation
The vis-viva equation gives you orbital speed at any point: v² = μ × (2/r − 1/a). Here r is the current distance from Earth's center, a is the semi-major axis. At perigee (r = a(1-e)) speed is maximum. At apogee (r = a(1+e)) speed is minimum. The ratio of perigee speed to apogee speed equals (1+e)/(1-e). For a Molniya orbit with e ≈ 0.74, that ratio is about 6.7. The math says everything you need: a satellite at the same total orbit can move at wildly different speeds depending on where it is.
Why Molniya orbits hover over Russia
A Molniya orbit has a 12-hour period and eccentricity around 0.74, with apogee over the northern hemisphere. Because Kepler's second law forces slow motion at apogee, the satellite spends about 8 of its 12 hours sitting nearly motionless over Russia (or Canada, depending on which half of the orbit you're on). It rushes through perigee on the other side of Earth in about 30 minutes. Soviet-era engineers used this to deliver broadcast and communications service to high latitudes where geostationary satellites are too low on the horizon to be usable.
When to use elliptical vs circular orbits
Circular orbits are simpler and give constant speed and altitude. They're ideal for global coverage constellations (GPS, Starlink, Iridium) where every satellite needs to behave identically. Elliptical orbits trade complexity for the ability to "hover" over a specific region for hours at a time, or to reach distant orbits cheaply (Hohmann transfers, lunar trajectories). The choice depends on what coverage pattern the mission needs.
Frequently asked questions
What is Kepler's second law?
Kepler's second law states that the line connecting a planet (or satellite) to its central body sweeps out equal areas in equal times. The mathematical consequence: orbital speed varies inversely with distance to the central body.
Why does a satellite move faster at perigee?
Conservation of angular momentum. Lower altitude means less moment arm, so the satellite must move faster to keep angular momentum constant. Physically: as the satellite falls toward Earth, gravity accelerates it; as it climbs away, gravity decelerates it.
What is a Molniya orbit?
A Molniya orbit is a 12-hour highly elliptical orbit (e ≈ 0.74) inclined at 63.4° with apogee over the northern hemisphere. Its design uses Kepler's second law to keep the satellite nearly stationary over high-latitude regions like Russia for about 8 hours per orbit.
What is the vis-viva equation?
The vis-viva equation gives orbital speed at any point in an orbit: v² = μ × (2/r − 1/a), where μ is the gravitational parameter, r is current distance from the central body, and a is the semi-major axis. It works for any conic section orbit.
What does eccentricity mean for orbital speed?
The higher the eccentricity, the bigger the speed difference between perigee (fastest) and apogee (slowest). The ratio is (1+e)/(1-e). A circular orbit (e=0) has constant speed; an e=0.5 orbit varies by a factor of 3; an e=0.74 Molniya orbit varies by a factor of about 7.
Related lessons
- Lesson 2: Altitude and Period — Higher orbits move slower. Drag the altitude slider and watch why.
- Lesson 7: Orbital Regimes — Four altitude bands. Four totally different design philosophies.
- Lesson 4: Ground Tracks and Inclination — Watch the path a satellite traces on Earth's surface. Inclination changes everything.
Open it in the simulator
Delta-V Academy is a free interactive orbital mechanics simulator that runs entirely in your browser. The 10-lesson curriculum covers everything from these basics through space domain awareness, with three difficulty levels (novice, intermediate, advanced) plus a kid-friendly mode. Launch the simulator and try Lesson 3 interactively.
See also: all FAQs · full curriculum · open the simulator