Measuring Rotation of Sun If the sun had a uniform surface (i.e., if there were no sunspots to look at), is there a practical way to measure its rotation?  In other words, if some external force flipped the sun's spin suddenly, could anyone on the planets notice?
My intuition is that general relativistic effects would cause the planetary orbits to change slightly, but I'm hoping someone could quantify that and see if it's detectable.  My guess is that this flip would cast some slight "effective drag force" slowing down planetary orbits.
EDIT: To avoid surface effects from the sun, let me simply assume that the "shell" of the sun is not spinning, but the underlying matter has all the spin.
 A: One can use the Doppler effect, which will shift spectral lines to the red at the side the rotates away from us and towards the blue on the side that rotates towards us. This is being used by astronomers who measure "rotational broadening" on stars which can not be resolved in telescopes. In that case it's all about measuring the rate of rotation, of course, and not the direction: https://en.wikipedia.org/wiki/Stellar_rotation
If we aren't allowed to use any kind of radiation (e.g. because we are rotating a black hole), then frame dragging will do: https://en.wikipedia.org/wiki/Frame-dragging. That, however, is an awfully hard experiment to do unless it's really a black hole and we are close, as the Gravity Probe B (https://en.wikipedia.org/wiki/Gravity_Probe_B) folks had to find out the hard way. The effect of the sun on the perihelion of Venus seems to be in the range of −0.0003 arcseconds/century... which is very hard to measure. 
A: You could try measuring the effects of the Lense-Thirring effect. It is an example of frame dragging. Essentially, an object that is orbiting near a massive object that is also rotating will have its axis undergo a change in its orientation.
There are two problems here:


*

*The rotating object must be large.

*The rotating object must not be rotating slowly.


Otherwise, the effects may be too small to accurately measure. Note that frame dragging was not used to calculate general relativistic effects in the precession of Mercury. For more information, see this answer.
The geodetic effect is another change in precession predicted by general relativity, caused by a body that is not necessarily rotating.
A: The shape of the Sun tells you something about it's mean rotation rate. The faster it spins, the more oblate it gets.
See  Why is the Sun almost perfectly spherical?
Of course you are not sensitive to the nuances of latitudinal or radial differential rotation. For the former you really do need to "see" the surface, for the latter you need helioseismology information, which yields rotational splitting of the pulsational modes. This would tell you about the internal rotation of the Sun, even if it were completely different to the surface rotation. This relies on being able to precisely measure velocities over the surface of the Sun.
