[en] The International Linear Collider (ILC) is an electron-positron-collider with a variable center-of-mass energy √(2) between 200 and 500 GeV. The small bunch sizes needed to reach the design luminosity of L_Peak=2.10³⁴ cm^-2s^-1 necessary for the physics goals of the ILC, cause the particles to radiate beamstrahlung during the bunch crossings. Beamstrahlung reduces the center-of-mass energy from its nominal value to the effective center-of-mass energy √(2'). The spectrum of the effective center-of-mass energy √(2') is the differential luminosity dL/d√(2'), which has to be known to precisely measure particle masses through threshold scans. The differential luminosity can be measured by using Bhabha events. The real differential luminosity is simulated by the GuineaPig software. The energy spectrum of the Bhabha events is measured by the detector and compared to the energy spectrum of Monte Carlo (MC) Bhabha events with a known differential luminosity given by an approximate parameterization. The parameterization is used to assign each MC event a weight. By re-weighting the events, until the energy spectra from the real and the MC Bhabha events match, the differential luminosity can be measured. The approximate parameterization of the differential luminosity is given by the Circe parameterization introduced by T. Ohl (1997), which does not include the correlation between the particle energies due to beamstrahlung. The Circe parameterization is extended to include the correlation and better describe the differential luminosity. With this new parameterization of the differential luminosity it is possible to predict the observed production cross section of a MC toy particle with a mass of 250 GeV/c² to a precision better than 0.2%. Using the re-weighting fit with the extended parameterization also allows the measurement of the beam energy spreads of σ_E=0.0014 for electrons and σ_E= 0.0010 for positrons with a precision of a few percent. The total error from the measured differential luminosity and beam energy spreads on the mass of a toy particle measured in a production threshold scan is found to be 7 MeV/c² for a 250 GeV/c² particle, with an integrated luminosity of 5fb^-1 per scanning point. (orig.)