In a little introductory book I have which I highly recommend for beginners, 'Relativity for the layman' by J.A. Coleman, here is a response...
The third possible explanation for the inability of the Michelson-Morley experiment to detect the ether assumed that the velocity of light was always constant with respect to the source which emitted it. This would mean that light always traveled at 186,000 miles a second with respect to the interferometer, regardless of how fast or slow it was moving with the earth through the ether. As a result, the velocity of light would vary with respect to the ether...
The main objection to this explanation was that it required velocity of light to vary with respect to the ether. This was contrary to the generally accepted notion of wave motion that the velocity of the wave must be constant in the material which carried the wave.... It was thus difficult for anyone really to believe that the velocity of light through ether was influenced by the velocity of the source...
There were also various astronomical observations which indicated that the velocity of light was independent of the velocity of the source. One of these was in connection with double stars. Double stars are two stars which are approximately the same size and are relatively close together. They rotate about each other with a fairly high velocity in somewhat the same way as would the ends of a dumb-bell... Now, some of these double stars rotate so that we are looking edgewise at the plane of rotation, i.e. we see one star coming towards us while the other is going away, and vice versa. If we assume that the velocity of the light leaving the star is increased or decreased by the velocity which the star is approaching or receding from us, then the star approaching us would appear to be rotating much faster than the receding one... The overall effect would be as if the stars were alternately speeding up and slowing down in their rotation about each other. Actual observation shows that this is not the case, however, and that the stars actually rotate about each other with uniform velocities. We conclude that it is entirely unlikely that the velocity of light is influenced by the velocity of the source, or that it is constant with respect to the source.