There is an ongoing effort to improve the sensitivity of the IceTop surface array at the South Pole by the addition of surface radio antennas. With a simulation study, we investigated the dependencies of the Akaryan relative to the geomagnetic emission in the band of 70-350 MHz at the South Pole. We used CoREAS to simulate the radio emission from cosmic ray air showers at that location using a star-shape pattern and used the polarization of the radio emission on the ground to calculate the relative Askaryan fraction at the Cherenkov ring of the radio footprint. We then investigated the dependence of the Askaryan fraction on several shower parameters, among them, the distance to the shower maximum ($d_{X_{\rm max}}$). Based on the finding, we have developed a model that uses the average fraction as well as the shower geometry to reconstruct the mass-sensitive depth of shower maximum ($X_{\rm max}$) for a given atmosphere. The theoretically achievable precision can compete with other methods for $X_{\rm max}$ measurement by radio arrays. Therefore, the Askaryan fraction - or more generally - the polarization of the radio signal may serve as an additional input for future multivariate approaches of $X_{\rm max}$ reconstruction.