What Is an Elliptical Orbit?
An elliptical orbit (or elliptic orbit) is a Kepler orbit with an eccentricity between zero and one. When eccentricity equals zero, the orbit is a perfect circle. The higher the eccentricity, the more elongated the ellipse — and the greater the difference between the farthest and closest points from the central body.
Every bound orbit around a massive body is an ellipse (Kepler's First Law). This includes the orbit of Earth around the Sun, the Moon around Earth, and every artificial satellite in low Earth orbit.
Ellipse Geometry: Semi-Major Axis, Semi-Minor Axis, and Eccentricity
An ellipse is described by two half-diameters:
- Semi-major axis (a) — half of the longest diameter. This is also the average orbital radius and determines the orbital period and energy.
- Semi-minor axis (b) — half of the shortest diameter. It describes how "round" or "flat" the ellipse is.
- Eccentricity (e) — a dimensionless number from 0 (circle) to <1 (ellipse): e = √(1 − b²/a²)
From the semi-major axis and eccentricity, we can find the apoapsis (farthest point) and periapsis (closest point) distances:
- ra = a(1 + e) — apoapsis distance
- rp = a(1 − e) — periapsis distance
For Earth's orbit around the Sun: a ≈ 149,597,870 km (1 AU), e ≈ 0.0167, so the aphelion is ≈ 152.1 million km and the perihelion is ≈ 147.1 million km.
Vis-viva Equation — Orbital Velocity at Any Point
The vis-viva equation gives the orbital speed of a body at any distance r from the central mass:
v = √( GM · (2/r − 1/a) )
Where:
- G = 6.674 × 10−11 N·m²/kg² — gravitational constant
- M — mass of the central body (star, planet)
- r — current distance from the central body
- a — semi-major axis of the orbit
Applying the vis-viva equation at apoapsis (r = ra) and periapsis (r = rp) gives the minimum and maximum orbital speeds. A satellite moves slowest at apoapsis and fastest at periapsis — this is a direct consequence of conservation of energy and angular momentum.
Kepler's Laws and the Orbital Period
Kepler's Third Law states that the square of the orbital period is proportional to the cube of the semi-major axis:
T = 2π · √( a³ / GM )
This means that knowing the semi-major axis and the mass of the central body is sufficient to determine the orbital period exactly.
Orbital Energy
The total mechanical energy of an orbit (kinetic + potential) is always constant and depends only on the semi-major axis:
ε = −GM / (2a) [J/kg — specific orbital energy]
E = −GMm / (2a) [J — total orbital energy]
Negative energy means the orbit is bound (the satellite cannot escape). The closer to zero, the weaker the gravitational binding.
Solar System Reference Table
| Planet | Semi-major axis (a) | Eccentricity (e) | Orbital period | Orbital speed (mean) |
|---|---|---|---|---|
| Mercury | 57.9 M km | 0.2056 | 87.97 days | 47.87 km/s |
| Venus | 108.2 M km | 0.0067 | 224.7 days | 35.02 km/s |
| Earth | 149.6 M km (1 AU) | 0.0167 | 365.25 days | 29.78 km/s |
| Mars | 227.9 M km | 0.0935 | 686.97 days | 24.07 km/s |
| Jupiter | 778.5 M km | 0.0489 | 11.86 yr | 13.07 km/s |
| Saturn | 1,432 M km | 0.0565 | 29.46 yr | 9.68 km/s |
| Uranus | 2,867 M km | 0.0457 | 84.01 yr | 6.80 km/s |
| Neptune | 4,515 M km | 0.0086 | 164.8 yr | 5.43 km/s |
| Moon (around Earth) | 384,400 km | 0.0549 | 27.32 days | 1.022 km/s |
Metric vs. American (Imperial) Units
This calculator supports both metric (km, km/s) and American (miles, mi/s) unit systems for all distance and velocity inputs and outputs. Masses are always given in solar masses (M☉) for the central body and Earth masses (M⊕) for the orbiting satellite — these units are standard in astrophysics regardless of the unit system chosen.
- 1 AU = 149,597,870.7 km = 92,955,807 miles
- 1 solar mass (M☉) = 1.989 × 1030 kg
- 1 Earth mass (M⊕) = 5.972 × 1024 kg