Is there an equation that describes the Sun's emitted power on the surface [in $\frac{W}{m^2}$] over a selected wavelength range (from $\lambda_1$ to $\lambda_2$) ?

I am guessing this can be calculated using Planck's law, but I just can't find the right equation to integrate it and get the result.



Could I integrate this equation from $\nu_1$ to $\nu_2$ intead of 0 to $\infty$:

$\frac{P}{A} = \frac{2 \pi h}{c^2} \int_0^\infty \frac{\nu^3}{ e^{\frac{h\nu}{kT}}-1} d\nu$

Like this:

$\frac{P}{A} = \frac{2 \pi h}{c^2} \int_{\nu_1}^{\nu_2} \frac{\nu^3}{ e^{\frac{h\nu}{kT}}-1} d\nu$

Where $\nu_1$ and $\nu_2$ are the frequencies of the light.

And then I would use Simpsons rule to numerically integrate and get the result?

And just a quick question, how inaccurate is Planck's law for the Sun at UV spectrum?


1 Answer 1


The Planck function is tricky to integrate. I found the following http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19680008986.pdf might help. In particular, Appendix C contains numerical tables of integrals for the Planck function between some frequency $\nu$ and infinity. Obviously, by subtracting one of these results using $\nu_1$ and another using $\nu_2$ will give the integral between $\nu_1$ and $\nu_2$.

It also contains the (short) Fortran code to do the integrals yourself.

The flux at the solar surface is $\pi B_{\nu}$ in W m$^{-2}$ Hz$^{-1}$ using the appropriate temperature for the solar photosphere.

NB: The Sun is not a blackbody to any significant precision near the peak of the distribution and instead you should integrate a spectrophotometric solar atlas or possibly a synthetic spectrum generated from a model atmosphere that is appropriately tuned for the Sun. Many such models exist - e.g. http://kurucz.harvard.edu/sun.html It is probably reasonable to treat the Sun as a blackbody from the near infrared and at longer wavelengths.

Here's a rough sketch of the intrinsic solar spectrum (and one after it has passed through the Earth's atmosphere), that I found at http://www.crisp.nus.edu.sg/~research/tutorial/optical.htm and which clearly shows the non-blackbody nature of the spectrum below 1 micron (including in the UV region).

enter image description here

  • $\begingroup$ Here is a related question about the sun being a bad blackbody (at least near the peak). $\endgroup$
    – Kyle Kanos
    Commented Nov 14, 2014 at 21:46
  • $\begingroup$ @KyleKanos Yes, this is good. I have added a picture for clarity. $\endgroup$
    – ProfRob
    Commented Nov 14, 2014 at 21:58
  • $\begingroup$ @RobJeffries Since I cannot figure out, does any of those models contain calculations for UV light? From around 200nm to 500nm. How would I go about integrating those? I am sorry if I am asking for to much, but I am lost and I really need your help. $\endgroup$ Commented Nov 15, 2014 at 10:54
  • $\begingroup$ @user1806687 I think there are UV solar atlases - the one I linked to stops at 300nm. You will need to do some research. How would you go about integrating them? Well you need to write some software that numerically integrates the data in the wavelength ranges you want. All this depends on how accurately you need the result... There are no ready-made solutions that I know of. $\endgroup$
    – ProfRob
    Commented Nov 15, 2014 at 12:04
  • $\begingroup$ @RobJeffries I just have no clue what which column means. $\endgroup$ Commented Nov 15, 2014 at 12:25

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