- P = period of Orbit
- t = time for position of interest
- T = time of periapsis passage
- e = eccentricity of the orbit

**True (Angle) Anomaly, φ**

*the angle of the object past periapsis with center the focus***Mean (Angle) Anomaly, M = 2π(t - T)/P**

*Actually the time past periapsis put in angle format***Eccentric (Angle) Anomaly, E**

*The angle of the object position projected on the reference circle.*

- Determine Eccentric Anomaly

**cos E = (e + cos φ)/(1 + e cos φ)** - Kepler's Equation

**M = E - e sin E**

t = (M/2π)P + T

**Example 1: For e = 0.3, φ = 90° find the time past periapsis**

cos E = (0.3 - 0)/1 + 0.3) = 0.2308: E = 76.66° = 1.338 rad

M = 1.338 - 0.3 sin(76.7°) = 1.046 rad

For a cirular orbit this would be 0.25 P

- Kepler's Equation
**M = E - e sin E**

Solve for E given M - use interative procedure. **cos(φ) = (cos E - e)/(1 - e cos E)**

**Example 2: For e = 0.3, (t - T)/P = 0.25, find the angle past periapsis**

M = 0.25 2π = 1.571

solve 1.571 = E - 0.3 sin E by interative use of the previous solution

E = M + e sin E

- 1st: E = 1.571 + 0.3 sin 90° = 1.871 rad = 107.2°
- 2nd: E = 1.571 + 0.3 sin 107.2° = 1.858 rad = 106.4°
- 3rd: E = 1.571 + 0.3 sin 106.4° = 1.859 rad = 106.5°
- 4th: E = 1.571 + 0.3 sin 106.5° = 1.859 rad

cosφ = [cos(106.5°)- 0.3]/[1 - 0.3 cos(106.5°)] = -0.455

For a cicular orbit this would be 90°

**Note:** All angles are in radians

1 radian = 360/2π = 57.3 degrees

June 10, 2005 - L.Bogan