True Anomaly Calculator
In orbital mechanics the true anomaly pins down where a body sits along its elliptical orbit, measured as an angle at the focus from the point of closest approach. It is related to the eccentric anomaly through one of Kepler's classic relations. This calculator takes the orbital eccentricity and the eccentric anomaly in degrees, then returns the true anomaly in degrees and radians along with the orbital radius expressed as a fraction of the semi-major axis. Every step uses the exact two-body Keplerian relations, no approximations or fitted figures.
True anomaly formula
factor = sqrt((1 + e) / (1 - e))
true anomaly = 2 * atan2(factor * sin(E/2), cos(E/2))
radius / a = 1 - e * cos(E)
Here e is the eccentricity and E is the eccentric anomaly in radians. Using the two-argument arctangent (atan2) of factor times sin(E/2) and cos(E/2) gives the true anomaly in the correct quadrant across the full orbit. The orbital radius as a fraction of the semi-major axis follows from r = a(1 - e cos E).
Orbital anomaly context
- The true anomaly is measured at the focus from periapsis to the body's current position.
- At a circular orbit (e = 0), true anomaly equals eccentric anomaly.
- The eccentric anomaly is usually found by solving Kepler's equation from the mean anomaly.
- The orbital radius is least at periapsis (E = 0) and greatest at apoapsis (E = 180 degrees).
- These relations are exact for the idealized two-body Keplerian orbit.
True anomaly: frequently asked questions
What is the true anomaly?
The true anomaly is the angle, measured at the focus of an elliptical orbit, between the direction of periapsis (closest approach) and the current position of the orbiting body. It tells you where the body is along its orbit relative to its closest point to the primary.
How is true anomaly found from eccentric anomaly?
The standard relation is tan(true anomaly / 2) equals the square root of (1 plus eccentricity) over (1 minus eccentricity), times tan(eccentric anomaly / 2). This calculator evaluates that exact Kepler relation from your eccentricity and eccentric anomaly.
What is the eccentric anomaly?
The eccentric anomaly is an auxiliary angle defined on the circle that circumscribes the orbital ellipse. It is typically obtained by solving Kepler's equation from the mean anomaly. Here you enter it directly so the tool can convert it to the true anomaly.
What does the orbital radius ratio mean?
The orbital radius at a given point equals the semi-major axis times (1 minus eccentricity times the cosine of the eccentric anomaly). This calculator reports that radius as a ratio of the semi-major axis, so you can scale it to any orbit size.
What eccentricity values are valid?
For a closed elliptical orbit the eccentricity lies between 0 (a circle) and just below 1. A value of 0 gives a true anomaly equal to the eccentric anomaly. As eccentricity approaches 1 the orbit becomes a very elongated ellipse approaching a parabola.
Official sources
- NASA: Basics of Space Flight, Orbital Mechanics.
- NASA Jet Propulsion Laboratory: Keplerian Elements and Anomalies.
Reviewed by the CalculatorHub team, edited by James Graham, 16 June 2026. See our methodology.