Jet breaks in gamma-ray burst (GRB) afterglows provide a direct probe of their collimation angle. Modeling a jet break requires an understanding of the jet spreading process, whereby the jet transitions from a collimated outflow into the spherical Sedov–Taylor solution at late times. Currently, direct numerical calculations are the most accurate way to capture the deceleration and spreading process, as analytical models have previously given inaccurate descriptions of the dynamics. Here (in paper I) we present a new, semi-analytical model built empirically by performing relativistic numerical jet calculations and inferring the relationship between the Lorentz factor, opening angle, and shock radius. We then use the analytical model to calculate the Lorentz factor and jet opening angle as a function of the shock radius and compare to the numerical solutions. Our analytic model provides efficient means of computing synthetic GRB afterglow light curves and spectra, which is the focus of paper II.
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