Peter Sutherland, dean of science and a professor of physics and astronomy at McMaster University, provides the following answer.
Image: FRANK BUB, University of New Hampshire
The equilibrium shape of a rotating star--or planet, for that matter--is not a sphere, but rather an flattened oblate spheroid. In other words, its diameter in the equatorial plane is longer than its diameter, pole to pole. The reason is that centrifugal effects are greatest at the rotational equator and work against the object's self-gravity.
Sir Isaac Newton first demonstrated this fact, calculating the difference between these two diameters for the earth. His calculation, though preliminary, was in approximate agreement with observation. [Note that the earth's substantial crust is solid, and so it does not necessarily reflect the equilibrium shape for today's rate of rotation; in the geological past, the earth was rotating more rapidly.]
How does this phenomenon effect the external gravitational field of a star or planet? A spherical gravitational source has an external gravitational field that falls off precisely as the inverse square of the distance from the sphere's center. An oblate spheroid adds another weaker component to the external field and so it falls off much more rapidly, inversely with the fourth power of the distance from the object's center.
For the earth, this modification to the gravitational field--along with many others that occur because our planet is not spherical--has important consequences for satellites and even the moon. Indeed, one long-term consequence is that the moon always presents the same face to the earth. (In this case, however, the earth's lack of sphericity results from the ocean tides, induced by the moon.)
When physicists apply the above considerations to the sun, there are implications for the Theory of General Relativity. By assuming that the sun was spherical, Albert Einstein (father of the Theory of General Relativity) explained with precision a variation in the orbit of Mercury, the so-called precession of the perihelion--a phenomenon that Newton's theory of gravity could not explain.
But in the 1960s, the American physicist Robert H. Dicke of Princeton, who died in 1997, noted that if the interior of the sun were rotating rapidly--compared to a slower speed observed at the surface--then the non-spherical component of the sun's gravitational field could produce up to 10 percent of the effect Einstein had computed, in which case, General Relativity would be an incomplete theory of gravity.
Ultimately, careful observations and analyses of the sun by Dicke, his students and colleagues could not confirm the idea. Thus, Dicke's conjecture and subsequent studies have tended to affirm Einstein's theory. And certainly, many other tests have also affirmed General Relativity--a subject reviewed in Clifford Will's book "Was Einstein Right?"