Generalization of the Gluckstern formulas II: Multiple scattering and non-zero dip angles

The first rules-of-thumb for the uncertainties in track momentum and direction of tracking detectors under inclusion of multiple scattering, as developed by Gluckstern [Nucl. Instr. and Meth. 24 (1963) 381] in the times of the bubble chamber, were limited to tracks with low curvature and equidistant measurement points with equal accuracy. The extension to strongly curved tracks with nonvanishing incident angle, arbitrary detector configuration and accuracy was published recently [M. Regler, R. Frühwirth, Nucl. Instr. and Meth. A 589 (2008) 109]. However, this extension is restricted to the (symmetry) plane with tracks with zero dip angle, perpendicular to the magnetic field, and does not treat multiple scattering. The present study extends the analytical approximate formulas for the calculation of uncertainties in track momentum and direction of “barrel” detectors to non-zero dip angles, including multiple scattering. The dip angle dependence of all terms of the error matrix is calculated. The results of a comparison with a linear least-squares fit are presented, showing excellent agreement. An open source implementation of the exact covariance matrices is described.