Carbon stars in the X-Shooter Spectral Library: II. Comparison with models

In a previous paper, we assembled a collection of medium-resolution spectra of 35 carbon stars, covering optical and near-infrared wavelengths from 400 to 2400 nm. The sample includes stars from the Milky Way and the Magellanic Clouds, with a variety of $(J-K_s)$ colors and pulsation properties. In the present paper, we compare these observations to a new set of high-resolution synthetic spectra, based on hydrostatic model atmospheres. We find that the broad-band colors and the molecular-band strengths measured by spectrophotometric indices match those of the models when $(J-K_s)$ is bluer than about 1.6, while the redder stars require either additional reddening or dust emission or both. Using a grid of models to fit the full observed spectra, we estimate the most likely atmospheric parameters $T_\mathrm{eff}$, $\log(g)$, $[\mathrm{Fe/H}]$ and C/O. These parameters derived independently in the optical and near-infrared are generally consistent when $(J-K_s)<1.6$. The temperatures found based on either wavelength range are typically within $\pm$100K of each other, and $\log(g)$ and $[\mathrm{Fe/H}]$ are consistent with the values expected for this sample. The reddest stars ($(J-K_s)$ $>$ 1.6) are divided into two families, characterized by the presence or absence of an absorption feature at 1.53\,$\mu$m, generally associated with HCN and C$_2$H$_2$. Stars from the first family begin to be more affected by circumstellar extinction. The parameters found using optical or near-infrared wavelengths are still compatible with each other, but the error bars become larger. In stars showing the 1.53\,$\mu$m feature, which are all large-amplitude variables, the effects of pulsation are strong and the spectra are poorly matched with hydrostatic models. For these, atmospheric parameters could not be derived reliably, and dynamical models are needed for proper interpretation.