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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?

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Helios 2 got the closest so far, at 0.29 AU. Solar Probe Plus will get much closer, if all goes well, down to 0.04 AU. Though only for a short time at each perihelion.

The fact that the Sun is extremely close to spherical, combined with the fact that the effect of even the first non-spherical gravitational parameter, $J_2$, falls off with the fourth power of distance, combined with the fact that you get fried when you get close, makes such measurements extremely difficult. This is further complicated, in a good way, by the fact that general relativity enters into the picture this deep in the Sun's gravitational field, so you need to measure other hard-to-measure parameters as well. Your experiment ends up being a General Relativity test.

The tracking of Mercury while MESSENGER was there improved our knowledge of the Sun's $J_2$ and put more stringent limits on the General Relativity parameters $\gamma$ and $\beta$.

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  • $\begingroup$ Not to mention, the delta-v requirements would be astronomical. $\endgroup$ – Aron May 11 '16 at 5:30
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To have such a probe is completeley unnecessary, since we have Helioseismology.
By observing the surface features of pressure waves that have a wavelength-dependent penetration depth into the sun itself, we can measure all the quantities you've just asked for.
Oblatenes, rotation (the one you've mentioned in your question), differential rotation, density and temperature profiles have all been obtained using this technique.

A famous result is the rotational profile of the sun, that I'm just gonna quote from Wikipedia: enter image description here
Here the rotational frequency $\Omega$ is plotted vs. discance from the stellar center $r$ in units of the solar radius $R$. The curves vary with the latitude as parameter and all join at $r/R \approx 0.7$ indicating that interior of this radius the sun rotates as solid body.

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