  We use CoREAS simulations to study the ratio of geomagnetic and Askaryan
radio emission from cosmic-ray air showers at the location of the South Pole.
The fraction of Askaryan emission relative to the total emission is determined
by the polarization of the radio signal at the time of its peak amplitude. We
find that the relative Askaryan fraction has a radial dependence increasing
with the distance from the shower axis -- with a plateau around the Cherenkov
ring. We further find that the Askaryan fraction depends on shower parameters
like zenith angle and the distance to the shower maximum. While these
dependencies are in agreement with earlier studies, they have not yet been
utilized to determine the depth of the shower maximum, $X_\mathrm{max}$, based
on the Askaryan fraction. Fitting these dependencies with a polynomial model,
we arrive at an alternative method to reconstruct $X_\mathrm{max}$ using a
measurement of the Askaryan fraction and shower geometry as input. Depending on
the measurement uncertainties of the Askaryan fraction, this method provides a
measurement of $X_{\mathrm max}$ with a resolution similar to other methods of
reconstructing $X_{\mathrm max}$ from radio observables. Although existing
template methods give superior resolution, they are more computationally
intensive. Consequently, the use of polarization to extract the Askaryan
fraction of the radio signal should be considered as an additional input
observable in future generations of template-fitting reconstruction and other
multivariate approaches for measuring $X_{\mathrm max}$.
