I'm new at understanding stellar classification and the spectral classification of stars. What is the exact reason TiO molecules (titanium oxide) dominate the spectrum for M dwarfs?
How did this TiO molecule form?
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There are two parts to the explanation.
The first is that Titanium and Oxygen exist as trace elements in the atmospheres of most Population I (metal-rich) stars in the disk of our Galaxy. If the molecule can form, then TiO is extremely opaque to infrared radiation over a variety of rotational and vibrational bands in the near-infrared.
The second is to do with the conditions in the photosphere. TiO has a dissociation energy of 6.9eV. That means that if there are enough photons in the radiation field surrounding the molecules that have energies exceeding this, then the molecule gets broken into its constituent atoms again.
To calculate the temperature below which TiO will form and be stable, you would need to do a full radiative transfer calculation in a stellar atmosphere, but the rough result should be obtained by inserting the relevant quantities (temperatures and densities) into a form of the Saha equation. The result is that TiO molecules would dissociate if temperatures exceeded 4000K.
However at much lower temperatures there are different molecules that would be the most stable that would tie up the available oxygen in the atmosphere (e.g. CO, VO, silicates etc.). So there is a window of temperatures (or spectral types) between about 4000K to 3000K (types M0-M5) where TiO features are dominant in the spectrum.