Caption: The normalized blackbody spectrum equivalent of the solar spectrum shown in 3 representations:

1. frequency representation (x_max = hν/(kT) = 2.821439372122078893... , F(x_max) ≅ 0.365).
2. log representation (x_max = hν/(kT) = 3.920690394872886343... , F(x_max) ≅ 0.592)
   (AKA fractional bandwidth).
3. wavelength representation (x_max = hν/(kT) = 4.965114231744276303... , F(x_max) ≅ 0.757).

Note, F(x_max) is the fractional integrated specific intensity from x = 0 to x = x_max.

The blackbody spectrum equivalent is the spectrum the Sun would have if it were a perfect blackbody radiator with a temperature exactly equal to the Sun's actual effective temperature (a characteristic photospheric temperature) which this plot sets to 5775 K which is slightly different from solar photosphere effective temperature = 5772 K (current best value).

Features:

1. Obviously, the shape of a spectrum depends on the representation, and so where a spectrum maximizes depends on the representation. The representation dependence of shape and maximum is clearly shown for the solar spectrum in the plot.

2. The 3 representations in the plot give maximizes at, respectively 502 nm = 0.502 μm (green band (fiducial range 0.495--0.570 μm), 635 nm = 0.635 μm (red (fiducial range 0.620--0.740 μm)), and 883 nm = 0.883 μm near-infrared (NIR, fiducial range 0.750--1.4 μm).

3. The energy distribution for the solar spectrum can be specified by percentiles for the wavelength representation: 10 % (< 0.380 μm), 25 % (< 0.502 μm), 50 % (< 0.711 μm), 75 % (< 1.065 μm), 90 % (< 1.624 μm), 90 % (< 1.624 μm), 100 % (< ∞).

4. We suggest that the log representation is the natural representation for all spectra.

   The argument:

   1. The differential relationship among the representations is

        
        dI = I_ν dν = νI_ν(dν/ν) = -λI_λ(dλ/λ) =  -I_λ dλ  ,

      where dI is differential integrated specific intensity, ν is frequency, I_ν is the specific intensity in the frequency representation, λ is wavelength, I_ν is specific intensity in the wavelength representation, and the minus sign is because increasing frequency dν > 0 causes corresponding decreasing wavelength dλ < 0.

      We see that νI_ν = λI_λ and both are what is called the log representation. The version νI_ν is plotted versus d㏑(ν) = dν/ν and the version λI_λ is plotted versus d㏑(λ) =dλ/λ.

      Since νI_ν = λI_λ, there is NO difference which of νI_ν and λI_λ you evaluate and plot, since plots of either are identical, except for mirror reflection about the either of the endpoints. These aspects of the log representation constitute the plotting convenience feature of the log representation.

   2. Is there direct way to see that dν/ν = - dλ/λ for electromagnetic radiation (EMR)? Yes. We take the differential of the natural logarithm of the phase velocity relation νλ=c and then follow obvious steps: i.e.,

        
        d㏑(c) = 0 = d[㏑(νλ)] = d[ ㏑(ν) + ㏑(λ) ] = d㏑(ν) + d㏑(λ)   
        d㏑(ν) = - d㏑(λ)  
          dν/ν = - dλ/λ  .  

   3. More important than the plotting convenience feature of the log representation is another feature.

      The size of spectrum structures in frequency and wavelength tend to be proportional to their characteristic absolute size. For example, Doppler shift and cosmological redshift both shift frequency/wavelength by common factors C for all frequency/wavelength Thus, the logarithmic size of the shift is

        
       ㏑(ν_shifted/ν) = ㏑(C) = -㏑(λ_shifted/λ)  

      throughout the spectrum.

      The from the above example and other cases, it follows that representation tends to give spectrum structures of equal importance relative their band equal importance in plots. This spectrum structure feature seems yours truly a great boon.

   UNDER CONSTRUCTION BELOW

5. In the log representation, blackbody spectrum specific intensity (i.e., Planck's law) is

                                            x**4
     νB_ν(T) = λB_λ(T) = 2c[(kT/(hc)]**4 ---------- ,
                                         [exp(x)-1]   

   where the Planck constant h = 6.62607015*10**(-34) J·s (exactly), the vacuum light speed c = 2.99792458*10**8 m/s (exactly) ≅ 3*10**8 m/s = 3*10**5 km/s ≅ 1 ft/ns, and the x = hν/(kT) =hc/(kTλ) is dimensionless (i.e., unitless) variable incorporating frequency and wavelength information, where the Boltzmann contant k = 1.380649*10**(-23) J/K (exactly) and T is temperature. For reference for the fundamental physical constants, see also NIST: Fundamental Physical Constants.