No, it would take millions of years before we noticed an effect. The reason is that the photons produced by nuclear reaction in the Sun's interor scatter very often. They travel only a very short distance inbetween collisions with the ions. After a collision they travel in some pretty much random direction compared to the direction before the collision. This is known as a random walk.
If the photons travel a distance d on average betwen collisions, then after N collisions they'll have travelled on average a distance of sqrt(N) d
(not N d)
d is about 10^(-6) meters. The radius of the Sun is of order 10^8 meters, so the number of collisions N needed for a photon to escape is of order 10^28. The total distance the photons will have travelled inbetween all the collsions is thus 10^22 meters, which is about a million lightyears.
If nuclear fusion in the sun's core suddenly stopped today, would the sky be dark in the daytime tomorrow? why
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