We present a study of carbon radio recombination lines towards Cassiopeia A using LOFAR observations in the frequency range 10-33 MHz. Individual carbon $alpha$ lines are detected in absorption against the continuum at frequencies as low as 16 MHz. Stacking several C$alpha$ lines we obtain detections in the 11-16 MHz range. These are the highest signal-to-noise measurements at these frequencies. The peak optical depth of the C$alpha$ lines changes considerably over the 11-33 MHz range with the peak optical depth decreasing from 4$times10^{-3}$ at 33 MHz to 2$times10^{-3}$ at 11 MHz, while the line width increases from 20 km s$^{-1}$ to 150 km s$^{-1}$. The combined change in peak optical depth and line width results in a roughly constant integrated optical depth. We interpret this as carbon atoms close to local thermodynamic equilibrium. In this work we focus on how the 11-33 MHz carbon radio recombination lines can be used to determine the gas physical conditions. We find that the ratio of the carbon radio recombination lines to that of the 158 $mu$m [CII] fine-structure line is a good thermometer, while the ratio between low frequency carbon radio recombination lines provides a good barometer. By combining the temperature and pressure constraints with those derived from the line width we are able to constrain the gas properties (electron temperature and density) and radiation field intensity. Given the 1$sigma$ uncertainties in our measurements these are; $T_{e}approx68$-$98$ K, $n_{e}approx0.02$-$0.035$ cm$^{-3}$ and $T_{r,100}approx1500$-$1650$ K. Despite challenging RFI and ionospheric conditions, our work demonstrates that observations of carbon radio recombination lines in the 10-33 MHz range can provide insight into the gas conditions.
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