Understanding True Anomaly
Mean, eccentric and true anomaly
Three related angles describe a satellite's progress around its orbit, and they are easy to confuse. All three are measured from perigee, all three read 0° at perigee and 180° at apogee, and for a perfectly circular orbit they are identical. Between those points they diverge — the more eccentric the orbit, the wider the gap. Only true anomaly is the real, physical angle to the satellite; the other two are computational stepping stones.
| Anomaly | Symbol | What it represents | Rate over time |
|---|---|---|---|
| Mean | M | A fictitious angle moving at a constant rate (a body on a same-period circular orbit) | Uniform / linear |
| Eccentric | E | A geometric auxiliary angle on the circle that circumscribes the ellipse | Non-uniform |
| True | ν, θ or f | The actual angle from perigee to the satellite, at Earth's centre | Non-uniform — fastest at perigee |
From time to position: Kepler's equation
Finding a satellite's true anomaly at a given moment takes three steps. First, advance the mean anomaly linearly with time using the mean motion (n): M = M₀ + n·Δt. Second, solve Kepler's equation M = E − e·sin E for the eccentric anomaly E — a transcendental equation with no closed-form solution, so it is solved iteratively, usually by Newton–Raphson. Third, convert E to true anomaly with tan(ν/2) = √((1 + e)/(1 − e)) · tan(E/2). This mean → eccentric → true chain is essentially what an SGP4 propagator runs each time it turns a TLE into a position on the map.
Why the speed varies — Kepler's second law
A satellite on an elliptical orbit does not travel at a constant speed. Kepler's second law states that the line joining it to Earth sweeps equal areas in equal times, so near perigee — where the radius is short — it must move quickly, while near apogee — where the radius is long — it slows right down. The difference between true and mean anomaly is called the equation of the centre; it grows with eccentricity, from under two degrees for a near-circular orbit to nearly 90 degrees for a highly elliptical one. This is exactly why a Molniya orbit can dwell for hours over the northern hemisphere near apogee, then whip through perigee in minutes.
True anomaly among the orbital elements
True anomaly is the sixth of the six classical Keplerian elements — and the only one that changes from moment to moment. The other five fix the orbit itself: semi-major axis (size), eccentricity (shape), inclination and right ascension of the ascending node (the plane's orientation), and argument of perigee (how the ellipse is turned within that plane). True anomaly then says where along that fixed ellipse the satellite currently sits. On near-circular orbits, where perigee is poorly defined, engineers often add the argument of perigee and true anomaly together as the argument of latitude (ω + ν), measured from the ascending node instead.