The Six Keplerian Elements, and What Each One Does

You need exactly six numbers to describe an orbit. Six, because you are replacing a 3D position and a 3D velocity, which is three numbers each.

The Keplerian elements are one way to write those six. What makes them useful is that five of them stay the same while the satellite moves. Only one of them changes with time.

Compare that to plain x, y, z coordinates, where all six numbers change every second. With Keplerian elements, five numbers describe the orbit and one number is the clock.

The shape: a and e

Semi-major axis a sets the size of the orbit. It also sets how long one lap takes:

T = 2π · sqrt(a³ / μ)

Here μ = GM⊕ = 398,600.4418 km³/s² for Earth. A 90-minute orbit means a ≈ 6,652 km, which is about 274 km above the ground (Earth's equatorial radius is 6,378 km).

Geostationary orbit is the value of a that makes T equal to one sidereal day. That is 86,164 seconds, not 86,400 — and the difference matters. It gives a = 42,164 km.

Eccentricity e says how far the orbit is from a circle. e = 0 is a perfect circle, and values between 0 and 1 are ellipses. At e = 1 the object escapes instead of orbiting.

Once you know a and e, you know the lowest and highest points:

r_perigee = a(1 − e)
r_apogee  = a(1 + e)

Most satellites in low Earth orbit have e < 0.01, which is close enough to a circle that you can treat the altitude as constant. Molniya orbits are the famous exception at e ≈ 0.74. They are stretched on purpose, so the satellite spends most of its time far away and moving slowly over high latitudes.

The orientation: i, Ω and ω

These three angles turn the orbit to face the right way. None of them changes the timing.

Inclination i is the tilt of the orbit compared to the equator. It decides which parts of the world the satellite flies over: a satellite at inclination i stays between latitude −i and +i and never goes outside that band.

This is the most important single number for a mission.

Sun-synchronous orbits sit near 98°. They use the fact that Earth is slightly fat at the equator, which slowly rotates the orbit by about 0.9856° per day. That is exactly the rate Earth moves around the Sun, so every pass happens at the same local time of day. This is why imaging satellites always see the same lighting.

Right ascension of the ascending node Ω says where the satellite crosses the equator heading north. Two satellites can have the same a, e and i but different Ω, which puts them in different planes. That is how you spread a constellation out.

Argument of perigee ω says where the lowest point sits inside the orbit. On a near-circular orbit this number barely means anything, because there is no clear lowest point. On a stretched orbit it decides which latitude you hover over. Molniya orbits fix ω = 270° to put the slow, far part of the orbit over the northern hemisphere.

The clock: ν

True anomaly ν is the only element that moves. It is the angle from the lowest point of the orbit to the satellite right now.

Turning a time into ν is the one awkward step. It goes through Kepler's equation, where M (mean anomaly) grows steadily with time and E (eccentric anomaly) is a helper angle:

M = E − e · sin(E)

There is no way to rearrange this to get E on its own. You have to solve it by repeated guessing, usually with Newton's method:

import math
 
 
def solve_kepler(M: float, e: float, tol: float = 1e-12) -> float:
    """Find E from M. Takes about 4 rounds for e < 0.7."""
    E = M if e < 0.8 else math.pi  # first guess
    for _ in range(50):
        delta = (E - e * math.sin(E) - M) / (1 - e * math.cos(E))
        E -= delta
        if abs(delta) < tol:
            return E
    raise RuntimeError(f"Kepler solver did not converge for M={M}, e={e}")
 
 
def true_anomaly(E: float, e: float) -> float:
    """Find the true anomaly from E. atan2 keeps the angle in the right half."""
    return 2 * math.atan2(
        math.sqrt(1 + e) * math.sin(E / 2),
        math.sqrt(1 - e) * math.cos(E / 2),
    )

Use atan2 in that second function, not atan. A plain atan cannot tell which half of the orbit you are in, so half your results come out on the wrong side.

Example: the ISS

Rounded numbers, close enough to work with:

From a: T = 2π · sqrt(6796³ / 398600.4) ≈ 5,575 s, which is 92.9 minutes.

From e: the lowest and highest points differ by 2ae ≈ 8 km. That is basically a circle.

From i: it never goes further north than 51.64°, which is why you cannot see it from Reykjavík.

Three numbers, and you already know the period, the shape and the coverage.

What this leaves out

Everything that makes real orbits drift. These six numbers assume Earth is a single point of gravity and nothing else exists. Real orbits also deal with:

• J2, the equatorial bulge. Earth is not a sphere. This is the biggest extra effect in low orbit, and it is what makes sun-synchronous orbits possible.
• Air drag. Below about 600 km there is still enough air to slow a satellite down and shrink its orbit. The ISS needs regular boosts for this reason.
• The Moon and the Sun, which pull on satellites in higher orbits.
• Sunlight, which pushes on large, light satellites.

This is why nobody uses plain Kepler maths to track a real satellite. They use SGP4, which includes a model of the bulge and a drag term. That is the subject of the next note.

These six numbers are still the right way to think about an orbit. They are just not the right way to predict one.

Part of the working notes behind SatLab.