Calculator Inputs
Example Data Table
These sample values show typical elliptical-orbit solutions using mean anomaly as the starting input.
| Case | Eccentricity (e) | Mean Anomaly (deg) | Eccentric Anomaly (deg) | True Anomaly (deg) | r / a |
|---|---|---|---|---|---|
| Low eccentricity orbit | 0.10 | 30.00 | 33.131579 | 36.407689 | 0.916258 |
| Moderate elliptic orbit | 0.30 | 75.00 | 92.176335 | 109.521956 | 1.011393 |
| High eccentricity orbit | 0.60 | 120.00 | 141.432267 | 160.154239 | 1.469123 |
| Very stretched ellipse | 0.80 | 210.00 | 196.772682 | 185.626639 | 1.765966 |
Formula Used
M = E - e sin(E)
Eₙ₊₁ = Eₙ - (Eₙ - e sin(Eₙ) - M) / (1 - e cos(Eₙ))
sin(E) = √(1 - e²) sin(ν) / (1 + e cos(ν))
cos(E) = (e + cos(ν)) / (1 + e cos(ν))
r = a(1 - e cos(E))
sin(ν) = √(1 - e²) sin(E) / (1 - e cos(E))
cos(ν) = (cos(E) - e) / (1 - e cos(E))
These equations apply to elliptical orbits. The calculator normalizes angles and reports both orbital geometry and convergence quality.
How to Use This Calculator
- Select the input mode that matches your known orbital quantity.
- Enter eccentricity and semi-major axis for the orbit.
- Provide mean anomaly, true anomaly, or radius as needed.
- Choose degrees or radians for angular values.
- Set tolerance and maximum iterations when solving from mean anomaly.
- Press the calculate button to display results above the form.
- Review the orbit graph, iteration log, and derived anomalies.
- Use the CSV or PDF buttons to export the computed summary.
FAQs
1. What is eccentric anomaly?
Eccentric anomaly is an auxiliary angle used to locate a body on an elliptical orbit. It connects orbital geometry with time through Kepler’s equation and makes many orbital calculations easier than working only with true anomaly.
2. How is eccentric anomaly different from mean anomaly?
Mean anomaly increases uniformly with time, while eccentric anomaly is a geometric angle on the auxiliary circle. Kepler’s equation links them, allowing time-based orbital motion to be converted into an ellipse position.
3. Why does this calculator use Newton-Raphson iteration?
Kepler’s equation usually cannot be rearranged into a simple closed-form solution for E. Newton-Raphson converges quickly for most elliptical orbits, especially when tolerance and iteration limits are chosen sensibly.
4. Which eccentricity values are supported?
This page supports elliptical orbits only, so eccentricity must stay from zero up to but not including one. Parabolic and hyperbolic trajectories require different anomaly definitions and different solving equations.
5. Can I solve eccentric anomaly from true anomaly directly?
Yes. The calculator includes a direct conversion mode using sine and cosine relations between ν and E. It then also reports the corresponding mean anomaly and orbital radius for the same orbit point.
6. Why does radius mode ask for an orbit branch?
A single radius can occur at two symmetric places on an ellipse, one above and one below the major axis. The branch selector chooses which eccentric anomaly solution should be reported.
7. Which units should I enter?
Choose degrees or radians before calculating. All entered angular values follow that choice, and the results section returns both the selected unit and radians for convenient checking.
8. What happens when eccentricity equals zero?
For a circular orbit, eccentric anomaly equals mean anomaly and also matches true anomaly. Radius mode becomes non-unique because the orbital radius stays constant at every position around the circle.