Final answer:
An observer on Earth would see an absorption spectrum with narrow, distinct lines due to a cool gas cloud absorbing specific wavelengths of the starlight. These lines are different from the star's own spectral lines and remain fixed in position.
Step-by-step explanation:
If a cool gas cloud is situated between a bright star and the Earth, you would expect to observe an absorption spectrum. This is because the cool gas can absorb certain wavelengths of light from the starlight passing through it, creating dark lines in the otherwise continuous spectrum. These lines, known as absorption lines, are indicative of the specific elements present in the intervening gas cloud.
The spectrum observed will have certain characteristic features. Firstly, compared to the broad spectral lines associated with the star itself, the lines from the narrow gas will be narrower and more distinct. Secondly, these narrow absorption lines will not shift in wavelength due to the Doppler effect as the binary star system does. Instead, they remain fixed because they are not part of the star's radial motion, signifying that they result from the gas between us and the star.