Orbital elements

Six numbers. That is all it takes to describe any orbit in the solar system, from the Moon’s to a comet’s. Learn to read them and you can pin down where any world will be, years ahead.

Six numbers describe any orbit in the solar system.

Look

To describe where a satellite or planet goes, you need to answer a few questions:

1. How big is the orbit?
2. What shape is it: round or stretched?
3. How tilted is it?
4. Which way is the tilt facing?
5. Which way does the stretched part point?
6. Where on the orbit is the body right now?

Each answer is one number. Together these six numbers are called the orbital elements, and they describe any orbit.

In the sandbox, the Molniya orbit is loaded. Russian communication satellites use this long, tilted orbit so they hang for hours high above the far north, where geostationary satellites are hard to see. Tick Side view to see the tilt, and play with the sliders.

Understand

Element	Symbol	What it sets
Semi-major axis	a	Size
Eccentricity	e	Shape
Inclination	i	Tilt of the orbital plane against a reference plane
Longitude of the ascending node	Ω	Direction in which the orbit crosses the reference plane going “up”
Argument of periapsis	ω	Where the closest point lies, measured within the orbital plane from the ascending node
Mean anomaly at epoch	M₀	Where the body is at a given moment (the epoch)

For planets, the reference plane is usually the ecliptic, the plane of Earth’s orbit, and the reference direction is the March equinox. For Earth satellites, the reference plane is Earth’s equator.

Some meaningful values:

• i = 0°: the orbit lies in the reference plane. A geostationary satellite has i = 0 and e = 0 around Earth’s equator.
• i = 90°: a polar orbit, passing over both poles.
• i > 90°: retrograde, going around backwards.
• The Molniya orbit uses i = 63.4° on purpose: at that tilt, Earth’s bulge does not make the periapsis drift around the orbit.

Master

The three angles are an ordered set of rotations. A point in the orbital plane (x′ towards periapsis, y′ 90° ahead) is carried into the reference frame by R_z(−Ω) · R_x(−i) · R_z(−ω). This is exactly the transformation in JPL’s “Approximate Positions of the Planets”, which the site’s orrery uses:

x = (cos ω cos Ω − sin ω sin Ω cos i) x′ + (−sin ω cos Ω − cos ω sin Ω cos i) y′

and similar expressions for y and z.

Planetary tables often give variants: the longitude of perihelion ϖ = Ω + ω (useful when i is small and Ω is poorly defined), and the mean longitude L = ϖ + M. The JPL tables list a, e, i, L, ϖ, Ω with their rates of change per century, because the other planets slowly perturb every orbit.

Why 63.4°? Earth’s equatorial bulge (the J₂ term) makes the argument of periapsis precess at a rate proportional to (5 cos² i − 1). It vanishes when cos² i = 1/5, i.e. i = 63.43° or 116.57°: the “critical inclinations”. Elements that stay constant in the two-body problem drift slowly in the real world; that is why element sets always come with an epoch, the moment they are valid for.