Because relativity is universal, our basic Doppler equations have to be capable of coping with everything, including moving gravitational fields and moving particulate media. Since everything has to use same Doppler formula, a planet that does not have an atmosphere has to have precisely the same equations of motion as one that does. Which would seem, at first sight, to mean that the effect on light of a moving refractive medium has no influence on a region’s light-geometry. Which is wrong.
An alternative suggestion is that since any curvature of light can be described using gravitational language, the atmosphere’s distortions should be considered to be part of the planet’s overall gravitational curvature, and the atmospheric effects on light are not additional or external to the planet’s gravitational description, but are already a part of it.
The light-dragging of individual particles or gas-flows in the atmosphere of a star has to be classifiable as gravitomagnetic. With the necessary non-SR equations, the individual gravitomagnetic attractions and repulsions don’t cancel out, but leave a net attraction and net redshift (SOD effect). So the thermal kinetic energy of the atmosphere contributes to the overall gravitational field strength as an additional gravitational field component.
A few details:
Trying to tackle even the concept of how the effect of a star’s atmosphere on the lightmetric (slowing-and-dragging) needs to connect with the effect of its gravity on the light-metric (slowing-and-dragging) tells us that we’re currently using the wrong equations.
If special relativity was correct for the Doppler shifts on individual atoms, treated as little SR observer-masses separated by vacuum, then SR would let us prove that the thermal energy of the atmosphere had no associated spacetime curvature. If the fusion processes of a star were to “switch off”, allowing it to radiate away its thermal energy as EM radiation and cool, E=mc2 requires the mass of the star, and its associated gravitational curvature, to reduce. But with the SR equations, the thermal activity of the “atmosphere” atoms, bouncing off each other, has zero net effect on lightbeam geometry, and makes no contribution to the star’s gravity.
The current system simply doesn’t work. Einstein built his system on the wrong geometry and the wrong equations.