Recover an elliptical position from elapsed time

1. Define the orbital evidence

Kepler’s equal-area law describes time, not uniform angular motion. An isolated two-body relative orbit is an ellipse with the attracting focus at one focus. The separation changes around an eccentric orbit, and the body moves faster near periapsis. A drawing with equally spaced angles therefore does not normally represent equally spaced observation times.

2. Connect geometry, force and conservation

Three angles have different roles. Mean anomaly M is a clock angle increasing uniformly from periapsis. Eccentric anomaly E parameterizes the ellipse through an auxiliary circle. The actual focus-centered direction, or true anomaly, follows atan2(y,x). The original model fixes semimajor axis and gravitational parameter to 1, solves E−e sin E=M, and constructs x=cos E−e and y=√(1−e²) sin E. These angles coincide only for a circle.

The separation is r=1−e cos E. Swept area from periapsis is √(1−e²)(E−e sin E)/2, so its fraction of the full ellipse area is M/(2π). Specific angular momentum is √(1−e²), and the vis-viva relation gives speed √(2/r−1). A position, an area fraction and a speed must be mutually consistent; a plausible-looking ellipse is not enough.

The original plotted points are equal-time calculations, not astrometric measurements. Bisection solves the monotone equation on one revolution; it does not fit a real ephemeris. Additional bodies, reference-frame motion, light travel time and measurement uncertainty would enter an actual orbit determination. At one complete revolution the elapsed area is a full ellipse even though the position returns to its starting point.

3. Work the original case

At e=0.6 and M=90°, one quarter of a period has elapsed. E≈119.8243°, position≈(−1.09734,0.69404), r≈1.29841 and speed≈0.735086. The area fraction is 0.25, not E/360. The original eight points mark equal time steps and crowd toward the slower part of the orbit.

4. Predict and investigate

Should equal elapsed times on an eccentric orbit give equal focus-centered angles or equal swept areas?

Hold eccentricity fixed and compare equal mean-anomaly steps. Reconstruct separation, swept-area fraction and speed, then compare with e=0. Identify an invariant that would reject a mistaken uniform-angle trajectory.

Source and boundaries

Primary mechanism reference. Checked 30 October 2026. Original prose, numerical cases and diagrams; source context is not a claim that these fictional records were observed. No actual mission operation or instrument aiming is requested.