Could one use a space probe to orbit the sun close enough to take measurements of angular / spherical gravitational field variations similar to what Juno plans to do for Jupiter?
Since the sun rotates on an axis, there would be some bulging of mass at the equator. And since density varies with the distance from the center of the sun, the effect of rotation might have varying effects depending on this distance which might affect the speed of a circumpolar orbiter.
How much insight into this density versus radius function can one obtain from gravitational measurements alone taken by such a circumpolar orbiter? I ask knowing that if the sun were not rotating and spherically symmetrical, then there would be no angular gravity field variations even though the density would still vary versus radius - at least from a Newtonian viewpoint (not sure if this would still be true from a GR standpoint).
Also, as a trivia question which space probe has the distinction of passing closest to the sun and survive? Messenger?