Least-Squares Orbit Fit: a Circle on a Kepler Ellipse

What you are seeing: noisy plane-of-sky positions sampled along an eccentric orbit, fit by least squares the way Gauss recovered the orbit of Ceres in 1801. The catch: fit a circle (cyan) to an eccentric orbit and the method converges cleanly to the wrong answer. The recovered radius stays biased no matter how many points you add, because the model, not the noise, is wrong; the residual RMS and the persistent centre offset show it.

Figure 1. Orbit fit; cyan = LS circle, orange = true orbit.

e (orbit)0.56

N samples20

σ noise0.05

WHAT TO TRY

Raise the eccentricity e: the least-squares circle fit gets steadily worse, since a circle cannot match an ellipse. The recovered radius is biased away from the true semi-major axis by an amount set by e, not by noise.
Add more samples N: the residual RMS shrinks but the bias does not, the recovered circle just converges more tightly to the wrong answer. More data cannot fix a misspecified model.
Increase the noise sigma: the scatter grows and the fit wobbles, but the systematic offset from fitting the wrong shape dominates once N is large. That is the lesson, a tight fit to the wrong model.