how much power would the entire earth radiate if it did not have an atmosphere
http://www.windows.ucar.edu/tour/link=/earth/climate/sun_radiation_at_earth.html
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.
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.
To calculate the total power radiated by the Earth without an atmosphere, you can use the Stefan-Boltzmann law. This law states that the power radiated by a perfect black body (which is a good approximation for the Earth) is proportional to the fourth power of its temperature.
1. Determine the effective temperature of the Earth's surface:
- The average temperature of the Earth's surface is approximately 288 Kelvin (15 degrees Celsius or 59 degrees Fahrenheit).
2. Calculate the power radiated by a perfect black body:
- The Stefan-Boltzmann constant is 5.67 × 10^-8 watts per square meter per Kelvin^4 (σ).
- The power radiated by a black body is given by the equation:
Power = σ * (temperature^4).
3. Substitute the values into the equation:
Power = 5.67 × 10^-8 * (288^4).
4. Calculate the result:
Using a calculator or a programming language, evaluate the expression:
Power ≈ 5.67 × 10^-8 * (288^4).
The result will be the total power radiated by the Earth without an atmosphere.
Note: This calculation assumes that the Earth's surface behaves like a perfect black body, which is an idealization. Additionally, it does not account for other factors such as solar radiation absorbed by the atmosphere and reflected back to space.