Time of the seasons

[cdeaglejr]

This HP Prime program can be used to determine the UTC calendar date and time of the equinoxes and solstices of the Earth. These events are the times when the apparent geocentric longitude of the Sun is an exact multiple of 90 degrees. This script uses Brent’s root-finder and a precision solar ephemeris to calculate these events.

Brent’s method requires an objective function that defines the nonlinear equation to be solved. The objective function for the spring and fall equinoxes is the geocentric declination of the Sun. The spring and fall equinoxes occur whenever the geocentric declination of the Sun is less than or equal to a user-defined convergence criterion.

For the summer and winter solstices, the objective function is theta - lambda, where theta = 90 degrees for the summer solstice, theta = 270 degrees for the winter solstice, and lambda is the geocentric longitude of the Sun. The summer and winter solstices occur whenever the difference delta = theta - lambda is less than or equal to a user-defined convergence criterion.

Brent’s method also requires an initial and final time which bounds the root of the objective function. The initial time for the spring equinox is March 15, for the summer solstice June 15, for the fall equinox September 15 and for the winter solstice December 15. For each event, the final time is equal to these initial dates plus 10 days.