$$\operatornamehav(\sigma) = \operatornamehav(\Delta\phi) + \cos\phi_1 \cos\phi_2 \operatornamehav(\Delta\lambda)$$ where $\operatornamehav(\theta) = \sin^2(\theta/2)$.

Spherical astronomy forms the basis for many practical problems in navigation, astrometry, and sky observation. The following sections present a selection of common problems, each with a detailed solution.

where GST is the Greenwich Sidereal Time, and longitude is the longitude of the observer.

And from that day on, Porto Astro had two navigators who spoke the language of spheres.

Determining the pass of satellites or the moon requires calculating rising, setting, and culmination times. 5. Summary Table of Solutions Problem Type Formula/Method Alt/Az →right arrow RA/Dec Spherical Law of Cosines RA/Dec →right arrow Alt/Az Spherical Law of Cosines Rise/Set Time Meridian Passage

She presented the first problem:

The Local Sidereal Time (LST) at any moment is directly related to the Right Ascension (RA) of a star crossing the local meridian. The fundamental relationship is:

M = E - e sin(E)

This is how ancient navigators determined latitude using Polaris (though Polaris is not exactly at the pole).