Left: ϕ angle after t = 100 for the Boris method and GCA, for particles of varying ﹩{qM}/m﹩ in a dipolar magnetic field. An analytic approximation ﹩\phi =2\pi \,(100/{T}_{\mathrm{dipole}})﹩ using Equation (70) is also shown. Right: difference in ϕ angles at t = 100 between the Boris method and GCA. The error in measuring ϕ for the Boris method is on the order of the gyroradius R_c, which is also indicated.