Suppose that the sun were to collapse from its current radius of about 700,000 km to a radius of only about 6000 km (about the radius of earth). what would you expect to happen as a result?

Only there will be a change in density of sun.

Explanation:

As your question says only the radius of sun is decreased and there is no change in the mass. So it means that the center of gravity will be the same and as a result the gravitational force remains the same. Only the density of sun changes as it shrinks.

Nothing significant.

Explanation: The energy flux emited by the sun will remain the same even if the sun's radius reduces drastically.

The energy from the sun reaching the Earth is given by the Stefan-Boltzmann equation.

Energy flux = Boltzmann's constant * T⁴

Energy flux = {{5.67*EXP(-8) watt per meter² per K⁴}*{5800K}⁴} = 63*EXP(6) Watt per meter²

Where EXP means 10^, and T= sun blackbody temperature 5800K.

It is good to know that the energy flux from the sun reaching the Earth does not depend on radius ( either it increases or decreases). The energy it supplies to earth remain the same.