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I understand that rocky bodies in the solar system all exhibit gravitational anomalies. Does the sun exhibit any gravitational anomalies? If so are they distinct enough that it effects the orbit of the Parker Solar Probe?

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The Sun rotates, taking about 24.5 days to complete a revolution at the equator, a bit longer at the poles. This makes the Sun have a bit of an equatorial bulge, which in turn means that the Sun's gravity field is not quite uniform. The Sun's J2 is very small and the effect is well known.

Very large stars are asymmetric due to being close to the Eddington limit. The Sun is too small of a star to have the kind of gravitational anomalies that very large stars and rocky planets exhibit.

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