Analytic Model for Off-Axis GRB Afterglow Images -- Geometry Measurement and Implications for Measuring $H_0$

We present an analytic model for measuring the jet core angle ($\theta_c$) and viewing angle ($\theta_{obs}$) of off-axis gamma-ray bursts independently of the jet angular structure outside of the core. We model the images of off-axis jets and using this model we show that $\theta_{obs}$ and $\theta_c$ can be measured using any two of the three following observables: the afterglow light curve, the flux-centroid motion, and the image width. The model is calibrated using 2D relativistic hydrodynamic simulations with a broad range of jet angular structures. We study the systematic errors due to the uncertainty in the jet structure and find that when using the light curve and centroid motion to determine $\theta_{obs}$ and $\theta_c$, our formulae can be accurate to a level of $5-10\%$ and $30\%$, respectively. In light of the Hubble tension, the systematic error in $\cos\theta_{obs}$ in GRBs originating in a binary compact object merger is of special interest. We find that the systematic uncertainty on the measurement of $\cos\theta_{obs}$ due to the unknown jet structure is smaller than $1.5\%$ for well-observed events. A similar error is expected if the microphysical parameters evolve at a level that is not easily detected by the light curve. Our result implies that this type of systematic uncertainty will not prevent measurement of $H_0$ to a level of $2\%$ with a sample of well-observed GW events with resolved afterglow image motion. Applying our model to the light curve and centroid motion observations of GW170817 we find $\theta_{obs}=19.2\pm 2~\deg$ (1$\sigma$) and $\theta_c=1.5-4~\deg$.