Orbital mechanics primer: TLEs and Keplerian elements

Every object in orbit can be described, at any moment, by six numbers: the Keplerian elements. This week introduces them through the most-used real-world format for distributing them: the two-line element set or TLE.

The six Keplerian elements

Two Body orbital mechanics gives us six independent parameters that uniquely identify an orbit's shape and orientation in inertial space:

1. Semi-major axis (a) — the orbit's "size" (half the longest diameter). For a circular orbit, this is the radius from Earth's center.
2. Eccentricity (e) — how oval the orbit is. e = 0 is circular; 0 < e < 1 is elliptical; e = 1 is parabolic (escape).
3. Inclination (i) — the tilt of the orbital plane relative to Earth's equator, in degrees. i = 0 is equatorial; i = 90 is polar; i > 90 is retrograde.
4. Right ascension of the ascending node (Ω) — where the orbit crosses the equator going north, measured from the vernal equinox direction.
5. Argument of periapsis (ω) — where the closest point to Earth lies, measured within the orbital plane from the ascending node.
6. True anomaly (ν) or mean anomaly (M) — where the satellite is in its orbit right now.

The first five describe the orbit's geometry; the sixth places the satellite on it at a specific time (the epoch).

The TLE format

A two-line element set is a NORAD-defined plain-text format that encodes a satellite's mean Keplerian elements plus a few drag/perturbation terms in 70 characters per line:

ISS (ZARYA)
1 25544U 98067A   24130.50145833  .00018539  00000-0  33188-3 0  9994
2 25544  51.6406 348.5395 0006703 117.9568 358.1729 15.50289267449420

Reading line 2 left to right: catalog number (25544), inclination (51.64°), RAAN (348.54°), eccentricity (0.0006703 — the leading 0. is implicit), argument of perigee (117.96°), mean anomaly (358.17°), mean motion (15.50289267 revs/day).

From TLE to orbital regime

The mean motion tells you the orbital period: period_minutes = 1440 / mean_motion. For the ISS at 15.50 revs/day, period is 92.9 minutes. From period and Kepler's third law you can back out the semi-major axis (~6,778 km from Earth's center, ~407 km altitude).

Combined with inclination, you can classify the orbit:

• LEO equatorial: i ≈ 0°, altitude < 2000 km. Rare; tropical launch sites only.
• LEO mid-inclination: i matches launch-site latitude. ISS (51.6° = Baikonur latitude).
• LEO sun-synchronous: i ≈ 98° (retrograde). Always passes the equator at the same local solar time. Used for Earth observation.
• MEO: 2,000–35,786 km. GPS (~20,200 km), GLONASS, Galileo.
• GEO: 35,786 km, i ≈ 0°. Period exactly 24 hours; appears stationary above the equator. Used by GOES, Himawari, and almost every communications satellite.
• Molniya: highly elliptical, i ≈ 63.4°. Used by Russia for high-latitude coverage where GEO doesn't reach.

Where TLEs come from

The U.S. Space Force's 18th Space Defense Squadron generates and publishes TLEs for every tracked object via Space-Track.org (account required) and the public mirror CelesTrak (no account, just curl). TLEs are updated daily and have an "epoch" timestamp — the further you propagate from the epoch, the more error accumulates. After ~1 week, accuracy degrades significantly; after ~30 days, refresh.

The lab fetches the ISS TLE from CelesTrak, parses it with sgp4.tle (or by hand using string slicing per the format spec), and prints each Keplerian element with its physical meaning. By the end you'll be able to look at any TLE and roughly visualize the orbit.