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Just to fill in some more infos:
Arpita Roy did some extensive work in her thesis about sky background rv contamination. Full hesis see here (chapter 2.4): https://etda.libraries.psu.edu/files/final_submissions/14948
The magnitude of the error introduced by lunar/solar background greatly depends on conditions (and target spectral type, RV shift between target + background etc.), but can easily exceed 10cm/s.
Worst case plot from thesis (for rv shift of target/background that causes maximum effect).
Our model is multiplicative. Do we want to consider sky emission, or lunar etc?
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