Question

how much power would the entire earth radiate if it did not have an atmosphere

Answer 1

http://www.windows.ucar.edu/tour/link=/earth/climate/sun_radiation_at_earth.html

Answer 2

If it did not have an atmosphere, its temperature would be different. That would have a greater effect (due to the T^4 term in the Stefan-Boltzmann equation) than the emissivity difference upon the amount of radiated energy.

But you also have to consider the effect of the atmosphere on the amount of absorbed energy from the sun.

The absorbed energy from the sun would probably be more, without clouds and ice reflecting a large fraction of the sun's radiation, as is the case now. The earth would therefore have to radiate more infrared to achieve a steady state, and this would require a hotter surface temperature, at least for the sunlit side.

Answer 3

Since you asked for how much power the earth would radiate, it would radiate more because of the lower albedo (i.e., more radiation is absorrbed from the sun). The albedo would be less and the solar absorbed power would be, roughly, 1.5 to 2 times higher. So would the emitted power. It is hard to say what the albedo would be with no atmosphere (or ocean), but the rocky planets with no atmosphere (Mars and Mercury), and the Moon) tend to be low albedo.