The Cosmic Background Radiation — Question 7

Question 7

The Rayleigh-Jeans approximation can be checked before calibrating a detector

At frequency nu=30 GHz and T=2.725 K use $x=h\nu/(k_BT)$, with $h/k_B=4.799\times10^{-11}$ K s. The exact-to-Rayleigh-Jeans intensity ratio is $x/(e^x-1)$.

1. Find x and this intensity ratio.

2. Find the percentage by which the Rayleigh-Jeans prediction exceeds the exact Planck intensity.

Question 7 - Solution

The Rayleigh-Jeans approximation can be checked before calibrating a detector

(a) Method and calculation

$$\boxed{x=(4.799\times10^{-11})(30\times10^9)/2.725=.528330}.$$$$\boxed{B_{\mathrm{Planck}}/B_{\mathrm{RJ}}=x/(e^x-1)\simeq.7590}.$$

(b) Interpretation and further calculation

$$\frac{B_{\mathrm{RJ}}-B_{\mathrm{Planck}}}{B_{\mathrm{Planck}}}=\frac{e^x-1}{x}-1.$$$$\boxed{\text{overestimate}\simeq31.8\%}.$$

Physical check

Low frequency must mean h nu much smaller than kT, not merely a frequency described informally as radio.