True Anomaly Calculator

Example data table

Values are rounded; your settings may produce slightly different results.

Professional guide to true anomaly

1) What true anomaly represents

True anomaly (ν) is the geometric angle, at the focus, between periapsis and the current radius vector. It is the most intuitive “where am I?” angle in an orbit because it links directly to position on the conic. In mission analysis it connects to lighting geometry, pointing constraints, ground track, and time-tagged state vectors.

2) Why multiple anomalies exist

Mean anomaly (M) increases nearly linearly with time for Keplerian motion, making it convenient for propagation. Eccentric anomaly (E) is an auxiliary angle for ellipses that simplifies the area-time relationship. Hyperbolic anomaly (H) plays the same role for escape trajectories. This calculator converts among these forms reliably.

3) Elliptical orbits and Kepler’s equation

For 0 ≤ e < 1, Kepler’s equation M = E − e sin(E) must be solved numerically except in special cases. Newton’s method converges quickly when tolerance and iteration limits are sensible. Once E is found, ν follows from a stable atan2 form that preserves the correct quadrant and behaves well near 0° and 180°.

4) Hyperbolic orbits and escape geometry

For e > 1, the hyperbolic form M = e sinh(H) − H is solved iteratively. Hyperbolic true anomaly is still meaningful as a geometric angle, but the radius grows rapidly. Small changes in H can produce large changes in ν at high eccentricity, so tightening tolerance helps when precision matters.

5) Radius, semi-major axis, and practical outputs

When semi-major axis a is supplied, the calculator also returns orbital radius r. For ellipses, r = a(1 − e cosE) highlights how periapsis distance shrinks as e rises. For hyperbolas, a is negative by convention; the calculator reports the internal sign choice and returns a positive radius magnitude.

6) Sensitivity, accuracy, and diagnostics

Iteration count and residual are reported so you can judge numerical health. A residual close to zero indicates the computed anomaly satisfies Kepler’s equation within tolerance. If convergence is slow, use a smaller tolerance, increase maximum iterations, or start with an anomaly-based mode when you already know E or H.

7) Common engineering and astronomy use cases

True anomaly supports burn timing, eclipse prediction, antenna pointing, and observation planning. In astrodynamics it is used with argument of periapsis and inclination to build the full state geometry. In observational work, ν helps map apparent motion along an orbit to expected brightness changes when distance and phase angle vary.

8) Data integrity and reporting workflow

For reproducibility, export results to CSV for spreadsheets or to PDF for reports and lab notebooks. Keep a consistent unit choice, document e and the chosen input mode, and record the solver tolerance used. This reduces ambiguity when you compare runs, validate an ephemeris, or share calculations with a team.

FAQs

1) What is the difference between mean anomaly and true anomaly?

Mean anomaly is a time-like parameter that advances almost linearly. True anomaly is the actual geometric angle locating the body on the orbit. Converting requires solving Kepler’s equation when e is not zero.

2) Why does the calculator wrap true anomaly into 0°–360°?

Wrapping provides a consistent, report-friendly angle range. It avoids negative angles and makes it easier to compare cases. Internally the solver still uses radians and stable atan2 forms to preserve the correct quadrant.

3) Can I compute radius without providing semi-major axis?

Not uniquely. Radius depends on the orbit’s size through a (or equivalently the semi-latus rectum p). If you provide a, the calculator can compute r from E or H; otherwise it reports only angular quantities.

4) What tolerance should I use for high eccentricity?

Start around 1e-10 for typical engineering work. If e is close to 1 or you need tight pointing accuracy, reduce tolerance and increase max iterations. Always check residual and iteration count for confidence.

5) Why is the hyperbolic semi-major axis shown as negative?

By standard convention, hyperbolic orbits use a negative semi-major axis. The radius equation uses that sign, but the physical radius is positive. The calculator applies the convention internally and reports a positive radius magnitude.

6) What does the “quadrant selection” mean in radius mode?

Radius alone can correspond to two symmetric true anomalies about periapsis. Quadrant selection lets you choose the near side (0°–180°) or far side (180°–360°). Pick the one consistent with your motion timeline.

7) Does this handle the parabolic case exactly (e = 1)?

Parabolic motion uses Barker’s equation and a different time parameter, so it is not solved directly from M here. If your orbit is nearly parabolic, use e slightly below or above 1, or use anomaly mode if available.