SPICE-Based Besselian Element Generator

Coefficients are shown in the traditional orientation: elements as columns, polynomial coefficients as rows. Classical elements appear first; extended elements z_☾ and z_☉ appear at the right.
Note that a (RA of the shadow axis) is not a traditional element; it provides an Earth-agnostic equatorial coordinate for the eclipse geometry, whereas μ (HA of the shadow axis) depends on terrestrial rotation and ΔT.

For ordinary Besselian calculations, use the traditional retained terms:
x and y usually use n = 0 through 3;
d, ℓ1 and ℓ2 usually use n = 0 through 2;
mu (or a) usually uses only n = 0 through 1;
tan f1 and tan f2 are usually treated as constants using n = 0 only.
The extended z☾ and z☉ values are diagnostic 3-D geometry terms rather than traditional path-calculation elements; use as many terms as needed for inspection or modeling.
Higher-order rows are shown for validation and experimentation, not because every displayed term is normally required.

The table and visualization below are evaluated from the selected SPICE-derived polynomial model. The Earth circle has radius 1 R⊕; ℓ1 and ℓ2 are shown to the same scale.

This optional section lets you paste published or legacy Besselian elements and compare them with the SPICE-derived polynomial elements generated above.