One of the evidence supporting the General Relativity is a shift of the apparent position of the stars during the total solar eclipse. I would like to know if the shift can be explained by other effects such as gradient of solar atmosphere. Is there an estimation of how much the solar atmosphere and other non-relativistic factors effect the observed shift ? And how different is the observed value of the shift from the theoretically predicted one ?
The effect of anomalous differential refraction (note that refraction in the atmosphere must be calibrated out by observing star fields at similar altitudes and in similar atmospheric conditions) in the Earth's atmosphere has been studied and concluded to be totally negligible (Lee & Bryant 1948).
The effects of refraction by the solar corona were dismissed in the original paper by Dyson, Eddington & Davidson (1919). The Sun's corona would need to have a density corresponding to about a hundredth of the Earth's atmosphere to provide comparable effects to GR light-bending, but it is in fact about 9 orders of magnitude lower.
I would like to know if the shift can be explained by other effects such as gradient of solar atmosphere.
Oh yes, there were no end of alternative explanations at the time. It should also be pointed out that there is an expected bending in Newtonian theory as well, but under GR the value is exactly double. There was also some controversy over the way certain measurements were simply ignored.
Unfortunately the majority of books don't bother even mentioning the alternatives; Gravitation is one of the few that mentions alternatives to anything (their discussion of tired light, for instance).
But you can often find works that are specifically historical that go into detail. Here's one for instance.
Hi koryakirp and welcome to PSE!
As you correctly pointed out, using light grazing the sun might expose the measurement to unwanted errors. Indeed the solar eclipses observations were always difficult, not least because they tended to occur (maximum time of totality) in such places as "jungles, middle of oceans, deserts and in arctic tundras" ( MTW page 1104). Total eclipses observations were carried out until 1968; now we use better techniques which confirm the theoretical result obtained by Einstein: 1.75 arc-seconds. The Newtonian theoretical deflection is half that value and all experiments , even the first ones, were clearly closer to the GR calculation.