I am interested in theoretical and practical considerations.
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3$\begingroup$ You mean what would happen if solar wind, magnetic field etc vanished? Because gravity-wise there would be no change. $\endgroup$– EelvexCommented Mar 29, 2011 at 5:02
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$\begingroup$ ok, what kind of doomsday device are you going to build? $\endgroup$– Tobias KienzlerCommented Mar 29, 2011 at 8:50
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1$\begingroup$ it would help if you clarify whether you are interested in higher order effects related to the Sun's non-spherical shape, as mentioned in Marek's comment on solomoan's answer. $\endgroup$– David ZCommented Mar 29, 2011 at 16:46
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It would be exactly the same, (atleast in Newtionian picture), no gravitational fields outside planets radius would change. The easiest way to see this I think is to use the gravitational analogoue of Gauss law.
Since we have spherical symmetry in both cases Int G dA = G*4*pi*r^2 ~ M
So G is constant.
See http://en.wikipedia.org/wiki/Gauss's_law
answered Mar 29, 2011 at 5:07
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1$\begingroup$ The equivalence of the gravitational fields would hold exactly in general relativity, too - because of Birkhoff's theorem, en.wikipedia.org/wiki/Birkhoff%27s_theorem_(relativity) - As long as there is no gravitating source "outside the Sun" or "outside the black hole", the outer portion of the solution has to be given by the Schwarzschild metric, and the mass parameter of its solution was guaranteed to be the same - and this mass is measured at infinity, too. So the changes inside the Sun are irrelevant outside it. $\endgroup$ Commented Mar 29, 2011 at 5:57
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2$\begingroup$ Both the answer and @Luboš's comment assume spherical symmetry of the Sun. Which isn't quite correct for this question: I think OP is aware that difference won't be big, so you should include also second order effects. The change would be in higher multipoles of gravitation (which are strictly zero for Schwarzschild BH but not for the Sun). $\endgroup$– MarekCommented Mar 29, 2011 at 6:36
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$\begingroup$ @Marek: perhaps, but you never know. It would have been nice if that were clarified in the question. $\endgroup$– David ZCommented Mar 29, 2011 at 16:45