Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 73 / 1 / vi

For Planck law expressed per unit frequency, put $x=h\nu /(k_{B}T)$. At fixed temperature, the frequency dependence is $B_{\nu }\propto T^3{x}^3/(e^x-1)$. Differentiation givesThe positive interior maximum therefore satisfies $3(1-e^{-x_{\ast }})=x_{\ast }$, with $x_{\ast }=2.821439\dots$ independent of temperature. The zero solution of the rearranged equation is a boundary, not the interior maximum. Hence the frequency form of Wien's displacement law isThis frequency peak is not $c$ divided by the peak wavelength of $B_{\lambda }$: the two spectral densities differ by the frequency-to-wavelength Jacobian.

Motion toward incoming cosmic microwave background radiation produces a blueshift. The photon occupation function of a blackbody depends on $h\nu /(k_{B}T)$, so the same Doppler factor multiplies both frequency and effective temperature. Looking forward along the Sun's peculiar velocity, one obtainsTaking the mean cosmic microwave background temperature as about $2.73\mathrm{K}$ givesThe numerical total inherits the rounded mean temperature; the robust estimate is the positive millikelvin dipole increment.