Orbital Mechanics Calculator

Eccentricity e describes the shape of an orbit. e = 0: perfect circle. e = 0.01-0.1: nearly circular (most LEO satellites). e = 0.5-0.8: highly elliptical (Molniya orbit). e approaching 1: extremely elongated (comets on long-period orbits). e = 1: parabolic escape. e > 1: hyperbolic (escape trajectory with excess velocity, like interplanetary spacecraft at planetary flyby).

The vis-viva equation v^2 = GM*(2/r - 1/a) gives the orbital speed at any point where r is the current distance from the center of the attracting body and a is the semi-major axis. It combines conservation of energy and angular momentum. For circular orbits (r = a), it simplifies to v = sqrt(GM/r). It is the most used equation in orbital mechanics for computing velocities.

Periapsis is the general term for the closest orbital point to the central body. Perigee is the periapsis of Earth-orbiting objects. Perihelion is the periapsis for Sun-orbiting objects. Perilune (or periselene) is for Moon-orbiting objects. Similarly: apoapsis (general), apogee (Earth), aphelion (Sun), apolune (Moon) for the farthest orbital point.

Mean motion n is the average angular velocity of the orbiting body: n = 360 / T (in degrees per minute or day). For Keplerian orbits, mean motion is constant. It is used in the two-line element (TLE) format that describes satellite orbits for tracking purposes. The ISS has a mean motion of about 15.5 revolutions per day.

Two-Line Elements (TLEs) are a standard format for describing Earth satellite orbits. They include the semi-major axis (encoded as mean motion), eccentricity, inclination, right ascension of ascending node, argument of perigee, and mean anomaly. TLEs are produced by NORAD/US Space Command and distributed freely for tracking purposes.

Atmospheric drag at low altitudes decelerates satellites, causing orbits to decay (shrinking a and period). The ISS orbit decays several kilometers per month due to residual atmosphere at 400 km. Regular reboosting maneuvers counteract this. Below about 120 km, drag causes rapid reentry within days or hours. Circular decay requires continuous reboost for long-duration missions.

A frozen orbit is one where the orbital elements (especially eccentricity and argument of perigee) remain nearly constant over time, despite perturbations from the Moon, Sun, and Earth's oblateness. Frozen orbits are preferred for Earth observation satellites because the spacecraft maintains a nearly constant altitude over its groundtrack, simplifying operations and instrument calibration.

Earth is not a perfect sphere — it bulges at the equator. The J2 term describes this oblateness. It causes two effects on satellite orbits: nodal precession (the orbital plane slowly rotates around Earth's polar axis) and apsidal precession (the orientation of the ellipse within the orbital plane rotates). Sun-synchronous satellites exploit nodal precession to keep the orbital plane aligned with the Sun.

A sun-synchronous orbit (SSO) is inclined to Earth's equator at about 97-99 degrees (slightly retrograde). Earth's J2 oblateness causes the orbital plane to precess eastward at about 0.9856 degrees/day — exactly matching Earth's revolution around the Sun. This keeps the orbital plane at a constant angle to the Sun-Earth line, so the satellite always crosses the equator at the same local solar time. Most Earth observation satellites use SSO for consistent lighting conditions.