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I have found information suggesting the difference between the surface and the center isn't even worth worrying about, but since we needed GR to account for Mercury's orbit, I can't help but wonder about more substantial differences.

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The GR corrections to Mercury's orbit are exceedingly tiny even though the mass of the sun is exceedingly large (on a human scale). The time dilation in the center of a star that is supporting fusion as determined by an outside observer at a safe distance can be calculated but it will be small. Wikipedia has a useful discussion of gravitational time dilation.

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