Thermodynamics – Why the Sun is Approximated as a Black Body at Approximately 5800 K

Question

Apparently spectral solar radiation is approximated by a black body at 5800 K. The spectral black body distribution (Planck distribution) is shown below (from Incropera, Fundamentals of Heat and Mass Transfer), with different temperatures including solar radiation at 5800 K.

1. The heat flux on an Earth surface perpendicular to sun rays (i.e., the solar constant), is roughly 1.36 kilowatts per square meter. Is this simply the integral of the above distribution?
2. Why is the sun approximated as a black body at 5800 K? Does this mean that the surface of the sun which is emitting radiation has a temperature of 5800 K? That seems kind of low.
3. Is solar radiation approximated as a black body at 5800 K only on Earth, or is it the same everywhere? Why don't atmospheric effects and scattering change measurements of the solar spectrum on Earth?

EDIT:
The solar constant is approximated by considering the Stefan-Boltzmann law (i.e., the integration of the spectral solar emission), the size of the sun, and the distance from the sun to Earth. A good derivation is shown here: https://www.youtube.com/watch?v=DQk04xqvVbU

Answer 1

1. Yes - the integral of the spectrum you refer to gives the total power per unit area emitted from the surface of the sun. If you multiply that by a factor of $\left(\frac{\text{Solar Radius}}{1\text{ AU}}\right)^2$ to account for the $1/r^2$ dependence of intensity on distance, then you'll get the solar constant you quote.
2. Yes. The sun is not at a single uniform temperature - the radiation which reaches Earth is mostly emitted from the photosphere (~6000 K) but the temperature varies dramatically between the different layers of the sun.
   

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3. Everywhere. The sun is very nearly an ideal black body. This is a property of the sun, not of a particular vantage point from which you're observing it. Furthermore, atmospheric effects dramatically change measurements of the solar spectrum on Earth.
   The upper atmosphere blocks nearly all of the radiation at higher frequencies than UV, and quite a lot of the IR spectrum is absorbed and scattered by greenhouse gases. Visible light passes through without much trouble (which is a substantial part of why we evolved to be sensitive to those frequencies) but the facts that the sky is blue and that sunsets are beautiful demonstrate that the atmosphere scatters visible light as well.

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