[swilhelm73]

Dr Duif also considered the possibility that, because the moon's shadow cools the air during an eclipse, this cooler, and thus denser, air might exert a different gravitational pull on the instruments.

Well, air, as all matter, does exert a gravitational pull, though an incredibly tiny one. And yes, cooler air, being denser, will exert slightly more pull.

But that being said, isn't the effect of concern here air resistance instead? The ambient temperature will have a much greater effect on air resistance then the gravitational pull exerted by said air...

[steve in DC]

A finite amount of air (or steel, or water, or whatever) has a given mass, no matter what temperature it may be.

The earth,s atmosphere is finite. Thus its mass is the same, no matter what its density might be.

Gravity is said to be inexorably tied to mass, not density. Giant gas planets have a gravitational pull that correlates with their mass, not their density. Although, that is not necessarily correct, in that their mass has been back calculated based upon the strength of their gravitational pull.