A formal verification requires integration of the two versions and. The area under each curve, that is, the total radiance, should be independent of variable. This signals that we should recheck are the differences in curve shape noted above. Equation is obtained, whether the variable is ν or λ. Heald (2003) has also discussed the issue of the ‘Wien peak’ position. This and a few more consequences of the non-linear transformation of the Planck distribution function have been described in more detail by Soffer and Lynch (1999). Consequently, in the wavelength representation the peak of solar radiation is in the middle of the sensitivity of the human eye, but in frequency space the maximum is outside our range of vision. The eye sensitivity curve is dimensionless and not affected by the transformation. Equation gives the corresponding maximum on the frequency axis at 341 THz, which corresponds to 0.88 μm, that is, well beyond 0.50 and actually outside the visible range. This wavelength almost agrees with the peak of the sensitivity of the human eye – but this agreement is only in the wavelength version. It agrees roughly with that for a blackbody at 5800 K. Comparing the diagrams above, we notice that the widths of the peaks increase with temperature in Fig. 8.2, while they decrease in Fig. 8.3.Īs an example we choose the solar spectrum, which is on the short wavelength side of Fig. 8.3. The non-linear transformation makes the corresponding infinitesimal steps unequal, which influences the shape of the curve. It gives the power density in each infinitesimal frequency or wavelength interval. Physically, it is a consequence of the Planck function being a distribution and having a dimension per frequency or per wavelength unit. ![]() The reason for this shift is the non-linear ν ⇔ λ coordinate transformation. Looking at Fig. 8.2 the maximum position of the 1000 K curve is considerably lower at ≈ 59 THz. ![]() ![]() If this wavelength is converted to frequency ν = c/ λ, we get 103 THz. The maximum of the 1000 K curve in Fig. 8.3 is ≈ 2.9 μm. A comparison of Figs 8.2 and 8.3, reveals, however, that the positions of the maxima are not conserved in the coordinate transformation.
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