Correct me if I’m wrong here. The thermosphere is hot due to its absorption of moderately high energy UV radiation. (<200nm) Cooler stars emit fewer high energy photons. So if the Earth orbited an orange dwarf, the earth would receive less moderately high energy UV radiation, causing the thermosphere to be much colder.
Additionally this would potentially allow helium to stay in the atmosphere for longer before escaping.
Is this correct?
The UV radiation that heats the thermosphere is actually not that highly energetic. Molecular oxygen is dissociated by UV radiation of 240nm already, which is only half the wavelength of blue light. Molecular nitrogen requires radiation of about 127nm which is about twice as energetic but you still could not call this high-energy UV radiation as it is not even able to ionize any elements, merely dissociate them.
Generally, less radiation in this wavelength region will obviously lead to less dissociation and thus less heating by the excess energy of the radiation (just look at Jupiter, which, due to its distance, is able to maintain an atmosphere of mainly (molecular) hydrogen), but the UV region of the spectrum of stars is not strictly coupled to the star's temperature as it is produced in regions of its atmosphere that are far from thermodynamic equilibrium, i.e. it is not given by a black body curve in this region. It differs by orders of magnitude from the Planck curve associated with the star's surface temperature and is in general difficult to model.
Having said this, recent observations appear to show that orange dwarfs have actually an increased output relatively to the Sun in the far UV as they get older (see https://www.sciencenews.org/article/orange-dwarf-stars-radiation-goldilocks-habitable-planets , or the original article at https://arxiv.org/pdf/2203.15237.pdf). This obviously would achieve the opposite to what you are suggesting.
