Observer sky math

Small-angle and low-order formulas on the celestial sphere: angular separation, chord length on a unit sphere, parallactic angle and airmass from RA/Dec at a JD, a rough Earth-orbital radial component toward a star, LSR correction with editable (U,V,W)_\\u2299, and solar twilight / golden-hour style times from the same Sun model used elsewhere (UT).

Not a substitute for observatory planning software: no refraction refinement, no topocentric Moon, no ephemeris-grade barycentric pipeline.