Orbital elements

An orbit is a trajectory in three-dimensional space. To pin it down completely takes six numbers — three for the orbit's size and shape, three for its orientation in space. These are the orbital elements, and they're the standard language astronomers and engineers use to talk about orbits. They're also exactly what's encoded in a TLE — the Two-Line Element set every orbiting object is described by.

Six numbers, three jobs

Counting why we need exactly six: an orbit's instantaneous state in 3D space is six numbers — three position coordinates and three velocity components. Newton's law of gravity then determines the entire future trajectory from that initial state. So six numbers' worth of information must specify the orbit. The orbital elements are just a different basis for the same six degrees of freedom — one chosen because each parameter has a clear geometric meaning instead of being some abstract Cartesian component.

The six split naturally into three jobs:

1. Two for the shape and size of the ellipse — semi-major axis $a$ and eccentricity $e$. These are the elements you already know from module 5.
2. Three for the orientation of the orbital plane in space — inclination $i$, longitude of ascending node $\Omega$, and argument of periapsis $\omega$. These are angles between the orbit and a fixed reference frame.
3. One for the timing — true anomaly $\nu$, or equivalently mean anomaly $M$ at some epoch. This says where on the orbit the body is right now.

• $a$ Semi-major axis. Sets the orbit's size — the average of periapsis and apoapsis. Determines period and total energy.
• $e$ Eccentricity. Sets the orbit's shape. Zero is circular; just under 1 is highly elongated; equal to 1 is parabolic escape.
• $i$ Inclination. The angle between the orbital plane and a reference plane (Earth's equator for satellites, the ecliptic for planets). Polar orbits have $i = 90°$; equatorial orbits have $i = 0°$.
• $\Omega$ Longitude of the ascending node. Where the orbital plane crosses the reference plane going north. Measured from a fixed direction in space (the vernal equinox for inertial frames).
• $\omega$ Argument of periapsis. The angle, in the orbital plane, from the ascending node to periapsis. Together with $\Omega$, this fixes where the long axis of the ellipse points.
• $\nu$ True anomaly. The angle from periapsis to the body's current position, measured at the focus. Position changes; the other five usually don't (or change only slowly).

The four reference directions

Three of the six elements are angles measured against a fixed reference frame. For Earth-orbiting objects the convention is the Earth-centered inertial (ECI) frame, with three axes that don't move with Earth's rotation:

• $+\hat x$: the vernal equinox direction — the direction from Earth's center toward the Sun on the first day of spring (technically, the ascending node of Earth's orbital plane through Earth's equatorial plane). This is the zero-point for $\Omega$.
• $+\hat z$: Earth's North celestial pole — perpendicular to Earth's equator.
• $+\hat y$: completes the right-handed coordinate system — toward the direction "ahead of" the vernal equinox by 6 hours of right ascension.

The reference plane is the equatorial plane (the $xy$ plane). The orbital plane is tilted away from it by $i$, hinged about the line of nodes (where the two planes intersect). The ascending node is the point on the orbit where the body crosses the equator going north; $\Omega$ is the angle from $+\hat x$ to that point, measured eastward in the equatorial plane.

Six elements, one picture

The diagram below puts all the geometry on a single illustration. The yellow disc is Earth's equatorial plane; the blue tilted disc is the orbital plane; they meet along the green dashed line of nodes. Within the orbital plane, the bright blue ellipse is the orbit itself, with periapsis marked in pink and the body's current position marked in white. Each of the six elements is labeled at its arc.

Try the elements

The interactive below renders a 3D view of an orbit with all six elements as sliders. The yellow disc is Earth's equatorial plane; the blue ellipse is the orbital plane. The orange arrow is the vernal equinox direction. Drag the sliders one at a time and watch which feature of the orbit changes.

Decoding a TLE

A two-line element set is the standard format for distributing orbital elements. CelesTrak publishes TLEs for tens of thousands of objects in Earth orbit, updated daily. They look like this:

Line 2 carries the orbital elements. From left to right after the satellite catalog number:

The semi-major axis isn't there directly — it's encoded in the mean motion via Kepler's third law. From $T = 2\pi\sqrt{a^3/GM_\oplus}$ and $n = 1/T$ in revolutions per second:

For the ISS at 15.50 rev/day = $1.794 \times 10^{-4}$ rev/s, this gives $a \approx 6{,}796\ \mathrm{km}$, or about 425 km of altitude — close to the ISS's actual cruising orbit. The mean anomaly $M$ converts to true anomaly $\nu$ via Kepler's equation (see the appendix). With those six numbers in hand you have the full orbit.

What's missing

The orbital elements above describe a perfect Keplerian orbit — one where Earth is a point mass and there are no other forces. Real Earth-orbiting satellites are perturbed by Earth's equatorial bulge (which causes nodal precession of $\Omega$ and apsidal precession of $\omega$ — both used deliberately in mission design), atmospheric drag at low altitude, solar radiation pressure, and gravity from the Sun and Moon. Real propagators like SGP4 (the one designed to consume TLEs) include simplified models of these perturbations. The "mean elements" in a TLE are tuned so that SGP4 reproduces the satellite's actual measured position, not so that they describe a perfect Keplerian orbit at face value.

Where this leads

The six orbital elements are the language of every satellite catalog, every mission plan, every TLE that SatView reads. The interactive above is essentially the same parameter set SatView uses to draw thousands of real satellites in real time. With the elements understood, the next module steps off the two-body problem into the three-body world — and the Lagrange points, where SOHO and JWST sit.