zero_eccentricity_exclusion

ell1fit.eccentricity.zero_eccentricity_exclusion(eps1, eps2)

Ask how firmly the joint posterior excludes a circular orbit.

The question “is the eccentricity significant?” is a question about the plane, not about either component: EPS1 and EPS2 are usually correlated, and each one alone can straddle zero while the pair together stays well away from the origin. So the statistic is the Mahalanobis distance of the origin from the sample cloud,

\[d^2 = \bar{\boldsymbol{\epsilon}}^{\mathsf{T}}\, \mathsf{C}^{-1}\,\bar{\boldsymbol{\epsilon}},\]

with \(\mathsf{C}\) the sample covariance. For a Gaussian posterior the credible region at level \(\alpha\) is the ellipse \(d^2 \le -2\ln(1-\alpha)\), so the origin lies exactly on the \(1 - e^{-d^2/2}\) contour: that number is returned as the credibility with which \(e = 0\) is excluded. EPS1 and EPS2 enter the phase model almost linearly at small \(e\), which is what makes the Gaussian step a fair one; a visibly banana-shaped corner plot is a sign it is not.

Returns:

credibilityfloat

Credible level at which the origin is excluded, in [0, 1).

sigmafloat

The same statement as an equivalent Gaussian significance – the number of standard deviations of a one-dimensional Gaussian carrying the same two-sided tail probability. 3.0 means “excluded with the credibility of a three-sigma result”, i.e. 99.73%.