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The graph of the Planck blackbody function has an interesting feature:## \\ ## ## \rho_o=\frac{\int\limits_{0}^{\lambda_{max}} L_{BB}(\lambda,T) \, d \lambda}{\int\limits_{0}^{+\infty} L_{BB}(\lambda, T) \, d \lambda} \approx .2500 ##,

where ## \lambda_{max} ##, in an exact derivation of Wien's law, obeys ## e^{-x}=1-\frac{x}{5} ##, where ## x \neq 0 ##, and ## x=\frac{hc}{\lambda_{max} k_B T} \approx 4.96 ##. From this we can very accurately compute ## \lambda_{max}## as a function of temperature with ## \lambda_{max}T=(\frac{hc}{k_B})(\frac{1}{X_o}) ## where ##X_o ## can be computed to any desired accuracy. The value of ## \rho_o ## can be shown to be independent of temperature ##T ##. ## \\ ## The value of ## \rho_o ## is so close to .2500, that in 1992 it took some calculus expansions long with some very high precision computing to show that the value of this fraction is not exactly equal to ## \frac{1}{4} ##, but instead approximately ## .2501 ##. (As I recall it came out to be ## \rho_o=.2500545...##). Today's very powerful computers could readily get a very accurate numerical number for this ratio to show that it is indeed ## \rho_o=.2500545... ##. Back in 1992, a couple of us made it a group effort to determine if ## \rho_o=\frac{1}{4} ##, or if ## \rho_o \approx .2500 ## , but not exact. After much effort, we concluded ## \rho_o=.2500545... ##.

where ## \lambda_{max} ##, in an exact derivation of Wien's law, obeys ## e^{-x}=1-\frac{x}{5} ##, where ## x \neq 0 ##, and ## x=\frac{hc}{\lambda_{max} k_B T} \approx 4.96 ##. From this we can very accurately compute ## \lambda_{max}## as a function of temperature with ## \lambda_{max}T=(\frac{hc}{k_B})(\frac{1}{X_o}) ## where ##X_o ## can be computed to any desired accuracy. The value of ## \rho_o ## can be shown to be independent of temperature ##T ##. ## \\ ## The value of ## \rho_o ## is so close to .2500, that in 1992 it took some calculus expansions long with some very high precision computing to show that the value of this fraction is not exactly equal to ## \frac{1}{4} ##, but instead approximately ## .2501 ##. (As I recall it came out to be ## \rho_o=.2500545...##). Today's very powerful computers could readily get a very accurate numerical number for this ratio to show that it is indeed ## \rho_o=.2500545... ##. Back in 1992, a couple of us made it a group effort to determine if ## \rho_o=\frac{1}{4} ##, or if ## \rho_o \approx .2500 ## , but not exact. After much effort, we concluded ## \rho_o=.2500545... ##.

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