[en] The effect of nonzero particle beam emittance and magnetic field errors on the on-axis spectral brightness (BT_o) of long undulators is discussed. It is shown that the quadratic dependence of BT_o on the number of undulator periods, N, is reduced by emittance to N^-x with 1 ≤x < 2. Further reductions in the intensity result from random magnetic field errors present in the undulator. An approximate model for the intensity of the central core of radiation of the principle harmonics is discussed and the results compared to those from Monte Carlo-type calculations where emittance is explicitly accounted for. An estimation of the effects of random field errors on the on-axis brightness in the presence of nonzero particle emittance is made. For the particular case of undulators proposed for the 7-GeV Advanced Photon Source, the results indicate that 5-m-long devices segmented into two sections will provide the required spectral intensity of a single long undulator when both emittance and magnet errors are considered